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THE BINARY STARS 




umaammmmam 

Plate I. The Thirty-Six-Inch Retractor of the Lick Observatory 



THE BINARY STARS 



BT ROBERT GRANT AITKEN 

Astronomer in the Lick Observatory 
University of California 



\ 



NEW YORK 
1918 




Copyright y igiSy by Douglas C. McMurtrte 



^3 

^3 




TO 

Sherburne Wesley Burnham 
THIS volume 

IS GRATEFULLY INSCRIBED 



PREFACE 

Credit has been given on many pages of this volume for 
assistance received in the course of its preparation; but I 
desire to express in this more formal manner my special grati- 
tude, first of all, to my colleague. Dr. J. H. Moore, for con- 
tributing the valuable chapter on The Radial Velocity of a 
Star; also to Director E. C. Pickering and Miss Annie J. 
Cannon, of the Harvard College Observatory, for generously 
permitting me to utilize data from the New Draper Catalogue 
of Stellar Spectra; to Professor H. N. Russell, of Princeton 
University, Professor F. R. Moulton, of the University of 
Chicago, Professor E. E. Barnard, of the Yerkes Observatory, 
and Dr. H. D. Curtis, my colleague, for putting at my disposal 
published and unpublished material; and, finally, to Director 
W. W. Campbell, for his constant interest in and encourage- 
ment of my work. Nearly all of the manuscript has been read 
by Dr. Campbell and by Dr. Curtis, and several of the chapters 
also by Dr. Moore and by Dr. Reynold K. Young, and I am 
deeply indebted to them for their friendly criticism. 

The book appears at this particular time in order that it 
may be included in the series of Semi-Centennial Publications 
issued by the University of California. 

R. G. AlTKEN 



December, igiy 



CONTENTS 

Page 

Introduction xiii 

Chapter I. Historical Sketch: The Early Period . . i 

Chapter II. Historical Sketch: The Modern Period 21 

Chapter III. Observing Methods, Visual Binary Stars 40 

Chapter IV. The Orbit of a Visual Binary Star . . 65 

Chapter V. The Radial Velocity of a Star, by Dr. 

J. H. Moore . 107 

Chapter VI. The Orbit of a Spectroscopic Binary Star 134 

Chapter VII. Eclipsing Binary Stars 167 

Chapter VIII. The Known Orbits of Visual and Spec- 
troscopic Binary Stars 192 

Chapter IX. Some Binary Systems of Special Interest 226 

Chapter X. A Statistical Study of the Visual Double 

Stars in the Northern Sky 252 

Chapter XI. The Origin of the Binary Stars .... 274 



ILLUSTRATIONS 

Page 
Plate I. The Thirty- Six-Inch Refractor of the Lick 

Observatory Frontispiece 

Plate II. The Micrometer for the Thirty-Six-Inch 

Refractor 41 

Plate III. Spectra of U4 Eridani, a Carinae, the Sun 

and az Centauri 114 

Plate IV. The Mills Spectrograph 117 

Plate V. Photographs of Krueger 60, in igo8 and in 

1915, by Barnard 230 

Figure i. Diagram to Illustrate Variable Radial 

Velocity 27 

Figure 2. Diagram to Illustrate Transit Method of 

Determining Micrometer-Screw Value . 44 

Figure 3. The Apparent Orbit of A 88 84 

Figure 4. The True and Apparent Orbits of a Double 

Star {after See) 91 

Figure 5. Apparent and True Orbits and Interpolating 
Curve of Observed Distances for a Binary 
System in which the Inclination is go° . 96 

Figure 6. Rectilinear Motion 104 

Figure 7. Diagram to Illustrate the Relations between 
Orbital Motion and Radial Velocity in a 
Spectroscopic Binary 135 

Figure 8. Velocity Curve of k Velorum 141 



Xll ILLUSTRATIONS 

Page 

Figure 9. King's Orbit Method. Graph for e = o.ys, 

0) = 60° 155 

Figure 10. Radial Velocity Curve for f Geminorum^ 

Showing Secondary Oscillation .... 163 

Figure ii. Light-Curve of the Principal Minimum of 

W Delphini 181 

Figure 12. The System of W Delphini 188 



INTRODUCTION 

It is the object of this volume to give a general account of 
our present knowledge of the binary stars, including such an 
exposition of the best observing methods and of approved 
methods of orbit computation as may make it a useful guide 
for those who wish to undertake the investigation of these 
systems; and to present some conclusions based upon the 
author's own researches during the past twenty years. 

The term binary star was first used by Sir William Herschel, 
in 1802, in his paper "On the Construction of the Universe," 
to designate "a real double star — the union of two stars, that 
are formed together in one system, by the laws of attraction." 

The term double star is of earlier origin ; its Greek equivalent 
was, in fact, used by Ptolemy to describe the appearance of v 
Sagittarii, two fifth magnitude stars whose angular separation 
is about 14', or a little less than half of the Moon's apparent 
diameter. It is still occasionally applied to this and other 
pairs of stars visible to the unaided eye, but is generally em- 
ployed to designate pairs separated by only a few seconds of 
arc and therefore visible as two stars only with the aid of a 
telescope. 

Not every double star is a binary system, for, since all of the 
stars are apparently mere points of light projected upon the 
surface of the celestial sphere, two unrelated stars may appear 
to be closely associated simply as the result of the laws of 
perspective. Herschel draws the distinction between the two 
classes of objects in the following words: 

" . . . if a certain star should be situated at any, per- 
haps immense, distance behind another, and but little deviating 
from the line in which we see the first, we should have the 
appearance of a double star. But these stars being totally 
unconnected would not form a binary system. If, on the con- 
trary, two stars should really be situated very near each other, 
and at the same time so far insulated as not to be materially 



xiv INTRODUCTION 

affected by neighboring stars, they will then compose a sep- 
arate system, and remain united by the bond of their mutual 
gravitation toward each other. This should be called a real 
double star." 

Within the last thirty years we have become acquainted 
with a class of binary systems which are not double stars in 
the ordinary sense of the term at all, for the two component 
stars are not separately visible in any telescope. These are 
the spectroscopic binary stars, so named because their existence 
is demonstrated by a slight periodic shifting to and fro of the 
lines in their spectra, which, as will be shown, is evidence of a 
periodic variation in the radial velocity (the velocity in the 
line of sight, toward or away from the observer) of the star. 
The only differences between the spectroscopic and the visual 
binary ("real double") stars are those which depend upon the 
degree of separation of the two components. The components 
of a spectroscopic binary, are, in general, less widely sep- 
arated than those of a visual binary, consequently they are 
not separately visible even with the most powerful telescopes 
and the systems have relatively short periods of revolution. 

In the present volume the two classes will be regarded as 
members of a single species. 



CHAPTER I 

HISTORICAL SKETCH: THE EARLY PERIOD 

The first double star was discovered about the year 1650 
by the Italian astronomer, Jean Baptiste Riccioli. This was 
f Ursae Majoris (Mizar). It is a remarkable coincidence that 
Mizar was also the first double star to be observed photographi- 
cally, measurable images being secured by G. P. Bond, at the 
Harvard College Observatory in 1857; and that its principal 
component was the first spectroscopic binary to be discovered, 
the announcement being made by E. C. Pickering in 1889. 

In 1656, Huyghens saw Ononis resolved into the three 
principal stars of the group which form the familiar Trape- 
zium, and, in 1664, Hooke noted that 7 Arietis consisted of 
two stars. At least two additional pairs, one of which proved 
to be of more than ordinary interest to astronomers, were dis- 
covered before the close of the Seventeenth Century. It is 
worthy of passing note that these were southern stars, not 
visible from European latitudes, — a Cruets, discovered by the 
Jesuit missionary, Father Fontenay, at the Cape of Good 
Hope, in 1685, and a Centauri, discovered by his confrere, 
Father Richaud, while observing a comet at Pondicherry, 
India, in December, 1689. 

These discoveries were all accidental, made in the course of 
observations taken for other purposes. This is true also of the 
double stars found in the first three-quarters of the Eighteenth 
Century. Among these were the discoveries of 7 Virginis, in 
1718, and of Castor, in 1719, by Bradley and Pound, and of 
61 Cygni, by Bradley, in 1753. 

No suspicion seems to have been entertained by these as- 
tronomers or by their contemporaries that the juxtaposition 
of the two star images in such pairs was other than optical, 
due to the chance positions of the Earth and the two stars in 
nearly a straight line. They were therefore regarded as mere 



2 THE BINARY STARS 

curiosities, and no effort was made to increase their number; 
nor were observations of the relative positions of the two com- 
ponents recorded except in descriptive terms. Father Feuille, 
for instance, on July 4, 1709, noted that the fainter star in the 
double, a Centauri, "is the more western and their distance is 
equal to the diameter of this star," and Bradley and Pound 
entered in their observing book, on March 30, 1719, that "the 
direction of the double star a of Gemini was so nearly parallel 
to a line through k and a of Gemini that, after many trials, we 
could scarce determine on which side of a the line from k par- 
allel to the line of their direction tended; if on either, it was 
towards /3." 

Halley's discovery, in 1718, that some of the brighter stars, 
Sirius, Arcturus, Aldebaran, were in motion, having unmis- 
takably changed their positions in the sky since the time of 
Ptolemy, unquestionably stimulated the interest of astron- 
omers in precise observations of the stars. These researches 
and their results, in turn, were probably largely responsible for 
the philosophical speculations which began to appear shortly 
after the middle of the Eighteenth Century as to the possi- 
bility of the existence of systems among the stars. Famous 
among the latter are the Cosmologische Briefe,^ published in 
1 761 by Lambert, in which it is maintained that the stars are 
suns and are accompanied by retinues of planets. Lambert, 
however, apparently did not connect his speculations with the 
double stars then known. Six years later, in 1767, John 
Michell, in a paper read before the Royal Society of London, 
presented a strong argument, based upon the theory of proba- 
bilities, that "such double stars, etc., as appear to consist of 
two or more stars placed near together, do really consist of 
stars placed near together, and under the influence of some 
general law, whenever the probability is very great, that there 
would not have been any such stars so near together, if all 
those that are not less bright than themselves had been scat- 
tered at random through the whole heavens." Michell thus 
has the credit of being the first to establish the probability of 

1 Cosmologische Briefe iiber die Einrichtung des Weltbaues, Ausgefertigt von J. H. Lam- 
bert, Augsburg, 1 76 1. 



THE BINARY STARS 3 

the existence of physical systems among the stars ; • but there 
were no observational data to support his deductions and they 
had no direct influence upon the progress of astronomy. 

The real beginning of double star astronomy dates from the 
activities of Christian Mayer and, in particular, of Sir William 
Herschel, in the last quarter of the Eighteenth Century. If 
a definite date is desired we may well follow Lewis in adopting 
the year 1779, for that year is marked by the appearance of 
Mayer's small book entitled "De novis in Coelo Sidereo Phae- 
nominis in miris Stellarum fixarum Comitibus," wherein he 
speculates upon the possibility of small suns revolving around 
larger ones, and by the beginning of Herschel's systematic 
search for double stars. 

The difference between Mayer's speculations and earlier ones 
is that his rest in some degree at least upon observations. 
These were made with an eight-foot Bird mural quadrant at 
Mannheim, in 1777 and 1778. At any rate, in his book just 
referred to, he publishes a long list of faint companions ob- 
served in the neighborhood of brighter stars.^ As one result 
of his observations he sent to Bode, at Berlin, the first collec- 
tion or catalogue of double stars ever published. The list 
contained earlier discoveries as well as his own and is printed 
in the Astronomisches Jahrbuch for the year 1784 (issued in 
1 781) under the caption, "Verzeichnis aller bisher entdeckten 
Doppelsterne." The following tabulation gives the first five 
entries: 





Gerade 

Aufst. 


Abwei- 
chung 


Unterschied 


Abstand 


Stellung 


Grosse 


in der 


in der 


des 
Klei- 








Aufst. 


Abw. 
















nern 




G. M. 


G. M. 


Sec. 


Sec. 


Sec. 




Andromeda beyde Qter 


8 38 


2945 N 


45 


24 


46 


S. W. 


Andromeda beyde Qter 


13 13 


20 18 N 


15 


29 


32 


S. 0. 


f Fische 6. und yter 


15 33 


625N 


22 


9 


24 


N. 0. 


beyAt Fische beyde yter 


19 24 


5 oN 





4 


4 


S. 


7 Widder beyde 5ter 


25 22 


18 13 N 


3 


12 


12 


s. w. 



* This list, rearranged according to constellations, was reprinted by Schjellerup in the 
journal Copernicus, vol. 3, p. 57, 1884. 



4 THE BINARY STARS 

In all, there are eighty entries, many of which, like Castor 
and 7 Virginis, are among the best known double stars. Others 
are too wide to be found even in Herschel's catalogues and a 
few cannot be identified with certainty. Southern pairs, like 
a Centauri, are of course not included, and curiously enough, 
6 Ononis is not listed. The relative positions given for the 
stars in each pair are little better than estimates, for precise 
measures were not practicable until the invention of the 
'revolving micrometer'. 

In his comments on Mayer's catalogue Bode points out that 
careful observations of such pairs might become of special 
value in the course of time for the discovery of proper motions, 
since it would be possible to recognize the fact of motion in 
one or the other star as soon as the distance between them had 
changed by a very few seconds of arc. Mayer himself seems 
to have had proper motions in view in making his observa- 
tions and catalogue rather than any idea of orbital motions. 

Sir William Herschel "began to look at the planets and the 
stars" in May, 1773; on March i, 1774, "he commenced his 
astronomical journal by noting that he had viewed Saturn's 
ring with a power of forty, appearing 'like two slender arms' 
and also 'the lucid spot in Orion's sword belt'." The earliest 
double star measure recorded in his first catalogue is that of 
6 Ononis, on November 11, 1776, and he made a few others 
in the two years following. It was not until 1779, however, 
that he set to work in earnest to search for these objects, for 
it was then that he conceived the idea of utilizing them to test 
a method of measuring stellar parallax suggested long before 
by Galileo. The principle involved is very simple. If two 
stars are in the same general direction from us and one is 
comparatively near us while the other is extremely distant, 
the annual revolution of the Earth about the Sun will produce 
a periodic variation in the relative positions of the two. As a 
first approximation, we may regard the more distant star as 
absolutely fixed and derive the parallax of the nearer one 
from the measured displacements. 

It seemed clear to Herschel that the objects best fitted for 
such an investigation were close double stars with components 



THE BINARY STARS 5 

of unequal brightness. He pointed out in his paper "On the 
Parallaxes of the Fixed Stars", read before the Royal Society 
in 1 78 1, that the displacement could be more easily and cer- 
tainly detected in a close double star than in a pair of stars 
more widely separated and also that in the former case the 
observations would be free from many errors necessarily af- 
fecting the measures in the latter. 

"As soon as I was fully satisfied," he continues, "that in the 
investigation of parallax the method of double stars would 
have many advantages above any other, it became necessary 
to look out for proper stars. This introduced a new series of 
observations. I resolved to examine every star in the heavens 
with the utmost attention and a very high power, that I 
might collect such materials for this research as would enable 
me to fix my observations upon those that would best answer 
my ends." 

In this reasoning, Herschel assumes that there is no physical 
connection between the components of such close double stars, 
— a fact upon which every writer on the history of double star 
astronomy has commented. This was not an oversight on his 
part, for at the close of his first catalogue of double stars he 
remarks, "I preferred that expression {i.e., double stars) to any 
other, such as Comes, Companion, or Satellite; because, in my 
opinion, it is much too soon to form any theories about small 
stars revolving round large ones, and I therefore thought it 
advisable carefully to avoid any expression that might convey 
that idea." 

Herschel's telescopes were more powerful than any earlier 
ones and with them he soon discovered a far larger number 
of double stars than he had anticipated. With characteristic 
thoroughness he nevertheless decided to carry out his plan of 
examining "every star in the heavens," and carefully recorded 
full details of all his observations. These included a general 
description of each pair and also estimates, or measures with 
the "revolving micrometer," or "lamp micrometer," both in- 
vented by himself, of the apparent distance between the two 
components and of the direction of the smaller star from the 
larger. The direction, or position angle, of the smaller star, 



6 THE BINARY STARS 

by his definition, was the angle at the larger star between the 
line joining the two stars and a line parallel to the celestial 
equator. The angle was always made less than 90°, the letters, 
w/, sf, sp, and np being added to designate the quadrant. His 
first catalogue, presented to the Royal Society on January 10, 
1782, contains 269 double stars, "227 of which, to my 
present knowledge, have not been noticed by any person." 
A second catalogue, containing 434 additional objects, was 
presented to the same society in 1784. The stars in these 
catalogues were divided into six classes according to angular 
separation. 

"In the first," he writes, "I have placed all those which 
require indeed a very superior telescope, the utmost clearness 
of air, and every other favorable circumstance to be seen at 
all, or well enough to judge of them. ... In the second 
class I have put all those that are proper for estimations by 
the eye or very delicate measures of the micrometer. . 
In the third class I have placed all those . . . that 
are more than five but less than 15" asunder; . . . The 
fourth, fifth, and sixth classes contain double stars that are 
from 15" to 30", from 30" to i' and from i' to 2' or more 
asunder." 

Class I, in the two catalogues, includes ninety-seven pairs, 
and contains such systems as r Ophiuchi, 8 Herculis, e Bootis, 
^ Ursae Majoris, 4 Aquarii, and f Cancri. In general, Herschel 
did not attempt micrometer measures of the distances of these 
pairs because the finest threads available for use in his micro- 
meters subtended an angle of more than i". The following 
extracts will show his method of estimating the distance in 
such cases and of recording the position angle, and also the 
care with which he described the appearance of each object. 
The dates of discovery, or of the first observation, here 
printed above the descriptions, are set in the margin at the 
left in the original. 

H. I. September 9, 1779 

e Bootis, Flamst. 36. Ad dextrum femur in perizomate. Double. Very 
unequal. L. reddish; 6". blue, or rather a faint lilac. A very beautiful object. 
The vacancy or black division between them, with 227 is ^ diameter of 



THE BINARY STARS 7 

5.; with 460, I yi diameter of L.; with 932, near 2 diameters of L,; with 
1,159, still farther; with 2,010 (extremely distinct), ^ diameters of L. 
These quantities are a mean of two years' observation. Position 31" 34' n 
preceding. 

H. 2. May 2, 1780 

i Ursae Majoris. Fl. 53. In dextro posteriore pede. Double. A little 
unequal. Both w [white] and very bright. The interval with 222 is ^ 
diameter of L.; with 227, i diameter of L; with 278, near \]/2 diameter of 
L. Position 53° 47' s following. 

Careful examination of the later history of the stars of 
Herschel's Class I shows that the majority had at discovery 
an angular separation of from 2" to 3.5"; a half dozen pairs 
as wide as 5" are included (one with the ms. remark, "Too far 
asunder for one of the first class"); and a number as close 
as or closer than \" , Seven of these stars do not appear 
in the great catalogue of Struve, but five of these have been 
recovered by later observers, leaving only two that cannot be 
identified. 

In passing judgment upon the accuracy, or the lack of it, in 
Herschel's measures of double stars, it is necessary to hold in 
mind the conditions under which he had to work. His reflec- 
tors (all of his own construction) were indeed far more powerful 
telescopes than any earlier ones, especially the "twenty-feet 
reflector," with mirror of eighteen and three-quarter inches 
aperture, and the great "forty-feet telescope," with its four-foot 
mirror. But these telescopes were unprovided with clock- 
work; in fact their mountings were of the alt-azimuth type. 
It was therefore necessary to move the telescope continuously 
in both coordinates to keep a star in the field of view and the 
correcting motions had to be particularly delicate when high- 
power eye-pieces, such as are necessary in the observation of 
close double stars, were employed. Add the crude forms of 
micrometers at his disposal, and it will appear that only an 
observer of extraordinary skill would be able to make measures 
of any value whatever. 

No further catalogues of double stars were published by 
Herschel until June 8, 1821, about a year before his death, 
when he presented to the newly founded Royal Astronomical 



8 THE BINARY STARS 

Society a final list of 145 new pairs, not arranged in classes, 
and, for the most part, without measures. 

After completing his second catalogue, in 1784, Herschel 
seems to have given relatively little attention to double stars 
until about the close of the century and, though he doubtless 
tested it fully, there is no mention of his parallax method in 
his published writings after the first paper on the subject. A 
thorough review of his double star discoveries which he insti- 
tuted about the year 1797 with careful measures, repeated in 
some cases on many nights in different years, revealed a 
remarkable change in the relative positions of the com- 
ponents in a number of double stars during the interval 
of nearly twenty years since their discovery, but this 
change was of such a character that it could not be produced 
by parallax. 

We have seen that, in 1782, Herschel considered the time 
not ripe for theorizing as to the possible revolution of small 
stars about larger ones. Probably no astronomer of his own 
or of any other age was endowed in a higher degree than 
Herschel with what has been termed the scientific imagination ; 
certainly no one ever more boldly speculated upon the deepest 
problems of sidereal astronomy ; but his speculations were the 
very opposite of guesswork, invariably they were the results of 
critical analyses of the data given by observation and were 
tested by further observations when possible. Michell, in 
1783, applied his earlier argument from the theory of probabili- 
ties to the double stars in Herschel's first catalogue and con- 
cluded that practically all of them were physical systems ; but 
it was not until July, 1802, that Herschel himself gave any 
intimation of holding similar views. On that date he presented 
to the Royal Society a paper entitled "Catalogue of 500 new 
Nebulae, nebulous Stars, planetary Nebulae, and Clusters of 
Stars; with Remarks on the Construction of the Heavens", in 
which he enumerates "the parts that enter into the construc- 
tion of the heavens" under twelve heads, the second being, 
"H. Of Binary sidereal Systems, or double Stars." In 
this section he gives the distinction between optical and 
binary systems quoted in my Introduction and argues as to 



THE BINARY STARS 9 

the possibility of systems of the latter type under the law of 
gravitation. 

On June 9, 1803, followed the great paper in which he gave 
the actual demonstration, on the basis of his measures, that 
certain double stars are true binary systems. This paper, the 
fundamental document in the physical theory of double stars, 
is entitled, "Account of the Changes that have happened, 
during the last Twenty-five Years, in the relative Situation of 
Double-stars; with an Investigation of the Cause to which 
they are owing." After pointing out that the actual existence 
of binary systems is not proved by the demonstration that 
such systems may exist, Herschel continues, "I shall therefore 
now proceed to give an account of a series of observations 
on double stars, comprehending a period of about twenty- 
five years which, if I am not mistaken, will go to prove, 
that many of them are not merely double in appearance, 
but must be allowed to be real binary combinations of two 
stars, intimately held together by the bonds of mutual 
attraction." 

Taking Castor as his first example, he shows that the change 
in the position of the components is real and not due to any 
error of observation. Then, by a masterly analysis of every 
possible combination of motions of the Sun and the compo- 
nents in this, and in five other systems, he proves that orbital 
motion is the simplest and most probable explanation in any 
one case, and the only reasonable one when all six are considered. 
His argument is convincing, his conclusion incontrovertible, 
and his paper, a year later, containing a list of fifty additional 
double stars, many of which had shown motion of a similar 
character, simply emphasizes it. 

This practically concluded Sir William Herschel's contribu- 
tions to double star astronomy, for his list of 145 new pairs, 
published in 1821, was based almost entirely upon observations 
made before 1802. In fact, little was done in this field by any 
one from 1804 until about 1 8 16. Sir John Herschel, in that 
year, decided to review and extend his father's work and had 
made some progress when Sir James South, who had indepen- 
dently formed similar plans, suggested that they cooperate. 



10 THE BINARY STARS 

The suggestion was adopted and the result was a catalogue of 
380 stars, based upon observations made in the years 1821 to 
1823 with South's five-foot and seven-foot refractors, of 3^" 
and 5" aperture respectively. These telescopes were mounted 
equatorially but were not provided with driving-clocks. They 
were, however, equipped with micrometers in which the par- 
allel threads were fine spider lines. The value of the catalogue 
was greatly increased by the inclusion of all of Sir William 
Herschel's measures, many of which had not before been 
published. 

Both of these astronomers devoted much attention to double 
stars in following years, working separately however. South 
with his refractors, Herschel with a twenty-foot reflector 
(eighteen-inch mirror) and later with the five-inch refractor 
which he had purchased from South. They not only remea- 
sured practically all of Sir William Herschel's double stars, 
some of them on many nights in different years, but they, and 
in particular Sir John Herschel, added a large number of new 
pairs. Indeed, so numerous were J. Herschel's discoveries and 
so faint were many of the stars that he deemed some apology 
necessary. He says, " . . .so long as no presumption a 
priori can be adduced why the most minute star in the heavens 
should not give us that very information respecting parallax, 
proper motion, and an infinity of other interesting points, 
which we are in search of, and yet may never obtain from its 
brighter rivals, the minuteness of an object is no reason for 
neglecting its examination. . . . But if small double stars 
are to be watched, it is first necessary that they should become 
known ; nor need we fear that the list will become overwhelm- 
ing. It will be curtailed at one end, by the rejection of un- 
interesting and uninstructive objects, at least as fast as it is 
increased on the other by new candidates." The prediction 
made in the closing sentence has not been verified; on the 
contrary, the tendency today is rather to include in the great 
reference catalogues every star ever called double, even those 
rejected later by their discoverers. 

The long series of measures and of discoveries of double stars 
by Herschel and South were of great value in themselves and 



THE BINARY STARS II 

perhaps of even greater value in the stimulus they gave to the 
observation of these objects by astronomers generally, and well 
merited the gold medals awarded to their authors by the Royal 
Astronomical Society. The measures, however, are now as- 
signed small weight on account of the relatively large errors of 
observation due to the conditions under which they were of 
necessity made; and of the thousands of new pairs very few 
indeed have as yet proved of interest as binaries. The great 
majority are too wide to give the slightest evidence of orbital 
motion in the course of a century. 

The true successor to Sir William Herschel, the man who 
made the next real advance in double star astronomy, an 
advance so great that it may indeed be said to introduce a new 
period in its history, was F. G. W. Struve. Wilhelm Struve 
became the director of the observatory at Dorpat, Russia, in 
1 813, and soon afterwards began measuring the differences in 
right ascension and in declination between the components of 
double stars with his transit instrument, the only instrument 
available. A little later he acquired a small equatorial, inferior 
to South's, with which he continued his work, and, in 1822, he 
published his "Catalogus 795 stellarum duplicium." This 
volume is interesting but calls for no special comment because 
Struve's great work did not really begin until two years later, 
in November, 1824, when he received the celebrated Fraun- 
hofer refractor. 

This telescope as an instrument for precise measurements 
was far superior to any previously constructed. The tube was 
thirteen feet long, the objective had an aperture of nine Paris 
inches,^ the mounting was equatorial and of very convenient 
form, and, best of all, was equipped with an excellent driving 
clock. So far as I am aware, this was the first telescope em- 
ployed in actual research to be provided with clock-work 
though Passement, in 1757, had "presented a telescope to the 
King [of France], so accurately driven by clock-work that it 
would follow a star all night long." A finder of two and one- 
half inches aperture and thirty inches focus, a full battery of 

2 This is Struve's own statement. Values ranging from qM to 9.9 inches (probably Eng- 
lish inches) are given by different authorities. 



12 THE BINARY STARS 

eye-pieces, and accurate and convenient micrometers com- 
pleted the equipment, over which Struve was pardonably en- 
thusiastic. After careful tests he concluded that "we may 
perhaps rank this enormous instrument with the most cele- 
brated of all reflectors, viz., Herschel's." 

Within four days after its arrival Struve had succeeded in 
erecting it in a temporary shelter and at once began the first 
part of his well-considered program of work. His object was 
the study of double stars as physical systems and so carefully 
had he considered all the requirements for such an investigation 
and so thorough, systematic, and skilful was the execution of 
his plans that his work has served as a model to all of his suc- 
cessors. His program had three divisions: the search for 
double stars; the accurate determination of their positions 
in the sky with the meridian circle as a basis for future 
investigations of their proper motions; and the measure- 
ment with the micrometer attached to the great telescope 
of the relative positions of the components of each pair to 
provide the basis for the study of motions within the 
system. 

The results are embodied in three great volumes, familiarly 
known to astronomers as the 'Catalogus Novus', the 'Posi- 
tiones Mediae', and the 'Mensurae Micrometricae'. The first 
contains the list of the double stars found in Struve's survey 
of the sky from the North Pole to —15° declination. For the 
purposes of this survey he divided the sky into zones from 
7>^° to 10° wide in declination and swept across each zone 
from north to south, examining with the main telescope all 
stars which were bright enough, in his estimation, to be visible 
in the finder at a distance of 20° from the full Moon. He con- 
sidered that these would include all stars of the eighth mag- 
nitude and the brighter ones of those between magnitudes 
eight and nine. Struve states that the telescope was so easy 
to manipulate and so excellent in its optical properties that 
he was able to examine 400 stars an hour; and he did, in fact, 
complete his survey, estimated to embrace the examination of 
120,000 stars, in 129 nights of actual work in the period from 
November, 1824, to February, 1827. 



THE BINARY STARS I3 

Since each star had to be chosen in the finder, then brought 
into the field of view of the large telescope, examined, and, if 
double, entered in the observing record, with a general descrip- 
tion, and an approximate position determined by circle read- 
ings, it is obvious that at the rate of 400 stars an hour, only 
a very few seconds could be devoted to the actual examination 
of each star. If not seen double, or suspiciously elongated at 
the first glance, it must, as a rule, have been passed over. 
Struve indeed definitely states that at the first instant of obser- 
vation it was generally possible to decide whether a star was 
single or double. This is in harmony with my own experience 
in similar work, but I have never been content to turn away 
from a star apparently single until satisfied that further exami- 
nation on that occasion was useless. As a matter of fact, later 
researches have shown that Struve overlooked many pairs 
within his limits of magnitude and angular separation, and 
hence easily within the power of his telescope; but even so 
the Catalogus Novus, with its short supplement, contains 3,112 
entries. In two instances a star is accidentally repeated with 
different numbers so that 3,110 separate systems are actually 
listed. Many of these had been seen by earlier observers and 
a few that had entirely escaped Struve's own search were in- 
cluded on the authority of Bessel or some other observer. 

Struve did not stop to make micrometer measures of his 
discoveries while engaged in his survey, and the Catalogus 
Novus therefore gives simply a rough classification of the pairs 
according to their estimated angular separation, with estimates 
of magnitude and approximate positions in the sky based on 
the equatorial circle readings. He rejected Herschel's Classes 
V and VI, taking 32" as his superior limit of distance and divid- 
ing the stars within this limit into four classes: (i) Those under 
4"; (2) those between 4" and 8"; (3) those between 8" and 16"; 
and (4) those between 16" and 32". Stars in the first class 
were further distinguished as of three grades by the use of the 
adjectives vicinae, pervicinae, and vicinissimae. The following 
lines will illustrate the form of the catalogue, the numbers in 
the last column indicating the stars that had been published 
in his prior catalogue of 795 pairs: 



14 



THE BINARY STARS 



Nume- 
rus 


Nomen 
Stellae 


A. R. 


Decl. 


Descriptio 


Num. 
C. P. 


I 




oh 0.0' 


+36° 15' 


II (8.9) (9) 




2 


Cephei 316 


—0.0 


+78 45 


I (6.7) (6.7), vicinae 




3 


AncIromedae3i 


-0.4 


+45 25 


II (7.8) (10) = H.II83 


I 


4 




-0.9 


+ 7 29 


II (9), Besseli mihi non 
inventa 




5 


34 Piscium 


— I.I 


-f-io 10 


III (6) (10), Etiam 
Besseli 





The Catalogus Novus, published in 1827, furnished the work- 
ing program on which Struve's other two great volumes were 
based, though the Positiones Mediae includes meridian circle 
measures made as early as 1822, and the Mensurae Microme- 
tricae some micrometer measures made in the years 1824 to 
1828. Micrometer work was not actively pushed until 1828 
and four-fifths of the 10,448 measures in the 'Mensurae' were 
made in the six years 1 828-1 833. The final measures for the 
volume were secured in 1835 and it was published in 1837. 
The meridian observations were not completed until 1843, and 
the Positiones Mediae appeared nine years later, in 1852. 

The latter volume does not specially concern us here for it 
is essentially a star catalogue, giving the accurate positions of 
the S (the symbol always used to designate Struve's double 
stars) stars for the epoch 1830.0. The Mensurae Microme- 
tricae, on the other hand, merits a more detailed description, 
for the measures within it hold in double star astronomy a 
position comparable to that of Bradley's meridian measures in 
our studies of stellar proper motions. They are fundamental. 
The book is monumental in form as well as in contents, mea- 
suring seventeen and one-half inches by eleven. It is, as Lewis 
remarks, not to be taken lightly, and its gravity is not lessened 
by the fact that the notes and the Introduction of 180 pages 
are written in Latin. Every serious student of double stars, 
however, should read this Introduction carefully. 

Looking first at the actual measures, we find the stars ar- 
ranged in eight classes. Class I of the Catalogus Novus being 
divided into three, to correspond to the grades previously 



THE BINARY STARS 



15 



defined by adjectives, and Classes III and IV, into two each. 
The upper Hmits of the eight classes, accordingly, are i, 2, 4, 
8, 1.2, 16, 24, and 32", respectively. The stars in each class 
are further distinguished according to magnitude, being 
graded as lucidae if both components of the pair are brighter 
than 8.5 magnitude, and reliquae if either component is fainter 
than this. 

Sir John Herschel had early proposed that the actual date 
of every double star measure be published and that it be given 
in years and the decimal of a year. About the year 1828 he 
further suggested that position angles be referred to the north 
pole instead of to the equator as origin and be counted through 
360°. This avoids the liability to mistakes pertaining to Sir 
William Herschel's method. Both suggestions were adopted 
by Struve and have been followed by all later observers. Gen- 
erally the date is recorded to three decimals, thus defining the 
day, but Struve gives only two. The position angle increases 
from North (0°) through East, or following (90°), South (180°), 
and West or preceding (270°). 

The heading of the first section, and the first entry under it 
will illustrate the arrangement of the measures in the Men- 
surae Micrometricae: 



DUPLICES LUCIDAE ORDINIS PRIMI 

Quarum distantiae inter o".oo et i".oo 



Epocha 


Amplif. 


Distant. 


Angulus 


Magnitudines 


2 


Cephei 316. 


a =0^0/0. 


5 =78« 45' 




Major— 6.2 flava; 


mi 


nor =6.6 certe j 


lavior 


1828.22 


600 


0.72' 


342.5° 


6.5.7 


1828.27 


600 


0.84 


343.4 


6.5.7 


1832.20 


600 


0.94 


339.3 


6,6 


1832.24 


480 


0.70 


337.5 


6,6.5 


1833.34 


800 


0.85 


344-8 


6.5,6.5m 


Medium 1830.85 




0.810 


341.50 





l6 THE BINARY STARS 

The Introduction contains descriptions of the plan of work, 
the instrument, and the methods of observing, and thorough 
discussions of the observations. The systems of magnitudes 
and of color notation, the division of the stars into classes by 
distance and magnitude, the proper and orbital motions de- 
tected, are among the topics treated. One who does not care 
to read the Latin original will find an excellent short summary 
in English in Lewis's volume on the Struve Double Stars pub- 
lished in 1906 as Volume LVI of the Memoirs of the Royal 
Astronomical Society of London. Three or four of Struve's 
general conclusions are still of current interest and importance. 
He concludes, for example, that the probable errors of his 
measures of distance are somewhat greater than those of his 
measures of position angle and that both increase with the 
angular separation of the components, with their faintness, 
and with the difference in their magnitudes. Modern observers 
note the same facts in the probable errors of their measures. 
In their precision, moreover, and in freedom from systematic 
errors, Struve's measures compare very favorably with the 
best modern ones. 

His observations of star colors show that when the two com- 
ponents of a pair are of about the same magnitude they are 
generally of the same color, and that the probability of color 
contrast increases with increasing difference in the brightness 
of the components, the fainter star being the bluer. Very few 
exceptions to these results have been noted by later observers. 

Finally, in connection with his discussion of the division of 
double stars into classes by distance, Struve argues, on the 
theory of probabilities, that practically all the pairs in his first 
three classes (distance under 4.00") and the great majority in 
his first five classes (distance less than 12") are true binary 
systems. With increasing angular separation he finds that the 
probability that optical systems will be included increases, 
especially among the pairs in which both components are as 
faint as, or fainter than 8.5 magnitude. This again is in har- 
mony with more recent investigations. 

The Russian government now called upon Struve to build 
and direct the new Imperial Observatory at Pulkowa. Here 



THE BINARY STARS I7 

the principal instruments were an excellent Repsold meridian 
circle and an equatorial telescope with an object glass of 
fifteen inches aperture. This was then the largest refractor 
in the world, as the nine-inch Dorpat telescope had been in 
1824. 

One of the first pieces of work undertaken with it was a re- 
survey of the northern half of the sky to include all stars as 
bright as the seventh magnitude. In all, about 17,000 stars 
were examined, and the work was completed in 109 nights of 
actual observing between the dates August 26, 1841, and 
December 7, 1842. The immediate object was the formation' 
of a list of all the brighter stars, with approximate positions, to 
serve as a working program for precise observations with the 
meridian circle. It was thought, however, that the more 
powerful telescope might reveal double stars which had escaped 
detection with the nine-inch either because of their small an- 
gular separation or because of the faintness of one component. 
This expectation was fully realized. The survey, which after 
the first month, was conducted by Wilhelm Struve's son. Otto, 
resulted in the discovery of 514 new pairs, a large percentage 
of which were close pairs. These, with Otto Struve's later 
discoveries which raised the total to 547, are known as the 02 
or Pulkowa double stars. The list of 514 was published in 
1843 without measures, and when, in 1850, a corrected cata- 
logue, with measures, was issued, 106 of the original 514 were 
omitted because not really double, or wider than the adopted 
distance limits, or for other reasons. But, as Hussey says, 
"it is difficult effectively to remove a star which has once 
appeared in the lists." Nearly all of the OS stars rejected 
because of wide separation have been measured by later ob- 
servers and are retained in Hussey's Catalogue of the OS Stars 
and in Burnham's General Catalogue. 

The early period of double star discovery ended with the 
appearance of the Pulkowa Catalogue. New double stars were 
indeed found by various observers as incidents in their regular 
observing which was mainly devoted to the double stars in the 
great catalogues which have been described and, in particular, 
to those in the 2 and the OS lists. The general feeling, how- 



l8 THE BINARY STARS 

ever, was that the Herschels and the Struves had practically 
completed the work of discovery. 

Many astronomers, in the half century from 1820 to 1870, 
devoted great energy to the accurate measurement of double 
stars; and the problem of deriving the elements of the orbit 
of a system from the data of observation also received much 
attention. This problem was solved as early as 1827, and new 
methods of solution have been proposed at intervals from that 
date to the present time. Some of these will be considered in- 
Chapter IV. 

One of the most notable of the earlier of these observers was 
the Rev. W. R. Dawes, who took up this work as early as 1830, 
using a three and eight- tenths inch refractor. Later, from 1839 
to 1844, he had the use of a seven-inch refractor at Mr, Bishop's 
observatory, and still later, at his own observatory, he installed 
first a six-inch Merz, then a seven and one-half inch Alvan 
Clark, and finally an eight and one-half inch Clark refractor. 
Mr. Dawes possessed remarkable keenness of vision, a quality 
which earned for him the sobriquet, 'the eagle-eyed', and, as 
Sir George Airy says, was also "distinguished . . . by a 
habitual, and (I may say) contemplative precision in the use 
of his instruments." His observations, which are to be found 
in the volumes of the Monthly Notices and the Memoirs of the 
Royal Astronomical Society, "have commanded a degree of 
respect which has not often been obtained by the productions 
of larger instruments." 

Another English observer whose work had great influence 
upon the progress of double star astronomy was Admiral W. H. 
Smythe, who also began his observing in 1830. His observa- 
tions were not in the same class with those of Dawes, but his 
Bedford Catalogue and his Cycle of Celestial Objects became 
justly popular for their descriptions of the double and multiple 
stars, nebulae, and clusters of which they treat, and are still 
"anything but dull reading." 

Far more important and comprehensive than that of any 
other astronomer of the earlier period after W. Struve was the 
double star work of Baron Ercole Dembowski who made his 
first measures at his private observatory near Naples in 1 851. 



THE BINARY STARS I9 

His telescope had an excellent object-glass, but its aperture 
was only five inches and the mounting had neither driving 
clock nor position circles. Nor was it equipped with a microm- 
eter for the measurement of position angles; these were de- 
rived from measures of distances made in two coordinates. 
With this instrument Dembowski made some 2,000 sets of 
measures of high quality in the course of eight years, though 
how he managed to accomplish it is well-nigh a mystery to 
observers accustomed to the refinements of modern microm- 
eters and telescope mountings. 

In 1859, he secured a seven-inch Merz refractor with circles, 
micrometer, and a good driving clock, and, in 1862, he resumed 
his double star observing with fresh enthusiasm. His general 
plan was to remeasure all of the double stars in the Dorpat 
and Pulkowa catalogues, repeating the measures in successive 
years for those stars in which changes were brought to light. 
His skill and industry enabled him, by the close of the year 
1878, to accumulate nearly 21,000 sets of measures, including 
measures of all of the S stars except sixty-four which for one 
reason or another were too difficult for his telescope. About 
3,000 of the measures pertain to the OS stars and about 1,700 
to stars discovered by Burnham and other observers. Each 
star was measured on several different nights and for the more 
interesting stars long series of measures extending over twelve 
or fifteen or even more years were secured. The comprehen- 
sive character of his program, the systematic way in which he 
carried it into execution, and the remarkable accuracy of his 
measures combine to make Dembowski's work one of the 
greatest contributions to double star astronomy. He died 
before his measures could be published in collected form, but 
they were later (i 883-1 884) edited and published by Otto 
Struve and Schiaparelli in two splendid quarto volumes which 
are as indispensable to the student of double stars as the Men- 
surae Micrometricae itself. 

Madler at Dorpat, Secchi at Rome, Bessel at Konigsberg, 
Knott at Cuckfield, Engelmann at Leipzig, Wilson and Gled- 
hill at Bermerside, and many other able astronomers published 
important series of double star measures in the period under 



20 THE BINARY STARS 

consideration. It is impossible to name them all here. Lewis, 
in his volume on the Struve Stars, and Burnham, in his General 
Catalogue of Double Stars, give full lists of the observers, the 
latter with complete references to the published measures. 



CHAPTER II . 

HISTORICAL SKETCH: THE MODERN PERIOD 

The feeling that the Herschels, South, and the Struves had 
practically exhausted the field of double star discovery, at 
least for astronomers in the northern hemisphere, continued 
for thirty years after the appearance of the Pulkowa Cata- 
logue in 1843. Nor were any new lines of investigation in 
double star astronomy developed during this period. Then, 
in 1873, a modest paper appeared in the Monthly Notices of 
the Royal Astronomical Society, entitled "Catalogue of Eighty- 
one Double Stars, Discovered with a six-inch Alvan Clark 
Refractor. By S. W. Burnham, Chicago, U. S. A." 

The date of the appearance of this paper may be taken as 
the beginning of the modern period of double star astronomy, 
for to Burnham belongs the great credit of being the first to 
demonstrate and utilize the full power of modern refracting 
telescopes in visual observations; and the forty years of his 
active career as an observer cover essentially all of the modern 
developments in binary star astronomy, including the dis- 
covery and observation of spectroscopic binaries, the demon- 
stration that the 'eclipsing' variable stars are binary systems, 
and the application of photographic methods to the measure- 
ment of visual double stars. 

Within a year after the appearance of his first catalogue 
Burnham had published two additional ones, raising the num- 
ber of his discoveries to 182. At this time he was not a profes- 
sional astronomer but an expert stenographer employed as 
official reporter in the United States Courts at Chicago. He 
had secured, in 1861, a three-inch telescope with alt-azimuth 
mounting, and, some years later, a three and three-quarter- 
inch refractor with equatorial mounting. "This was just good 
enough," he tells us, "to be of some use, and poor enough . . . 
to make something better more desirable than ever." In 1870, 



22 THE BINARY STARS 

accordingly, he purchased the six-inch refractor from Alvan 
Clark and erected it in a small observatory at his home in 
Chicago. With this instrumental equipment and an astronom- 
ical library consisting principally of a copy of the first edition 
of Webb's Celestial Objects for Common Telescopes, Mr. Burn- 
ham began his career as a student of double stars. His first 
new pair (jS 40) was found on April 27, 1870. 

The six-inch telescope, which his work so soon made fam'ous, 
was not at first provided with a micrometer and his earliest 
list of discoveries was printed without measures. Later, posi- 
tion angles were measured but the distances continued to be 
estimated. This lack of measures by him was covered to 
a considerable extent by the measures of Dembowski and 
Asaph Hall. 

Burnham's later career has been unique. He has held posi- 
tions in four observatories, the Dearborn, the Washburn, the 
Lick, and the Yerkes, and has discovered double stars also 
with the twenty-six-inch refractor at the United States Naval 
observatory, the sixteen-inch refractor of the Warner observa- 
tory, and the nine and four-tenths-inch refractor at the Dart- 
mouth College observatory. In all, he has discovered about 
1,340 new double stars and has made many thousands of 
measures which are of inestimable value because of their great 
accuracy and because of the care with which he prepared his 
observing programs. And yet, except for the two short periods 
spent respectively at Madison and at Mount Hamilton, he 
continued his work as Clerk of the United States District 
Court of Chicago until about eight years ago! He retired 
from the Yerkes observatory in 191 2. 

Burnham's plan in searching for new double stars was very 
different from that followed by his great predecessors. He 
did not attempt a systematic survey of the sky but examined 
the stars in a more random way. In his earlier work, while 
identifying the objects described in Webb's book, he made a 
practice of examining the other stars near them. Later, 
whenever he measured a double star, he continued this prac- 
tice, examining in this manner probably the great majority 
of the naked eye and brighter telescopic stars visible from our 



THE BINARY STARS 23 

latitudes. Many of the double stars he discovered with the six- 
inch refractor are difficult objects to measure with an aperture 
of thirty-six inches. They include objects of two classes almost 
unrepresented in the earlier catalogues: pairs in which the 
components are separated by distances as small as 0.2", and 
pairs in which one component is extremely faint, and close to 
a bright primary. In his first two lists he set his limit at 10", 
but later generally rejected pairs wider than 5". Jhe result is 
that the percentage of very close pairs, and therefore of pairs 
in comparatively rapid orbital motion, is far higher in his 
catalogue than in any'^of the earlier ones. 

Burnham's work introduced the modern era of double star 
discovery, the end of which is not yet in sight. No less dis- 
tinguished an authority than the late Rev. T. W. Webb, in 
congratulating Burnham upon his work in 1873, warned him 
that he could not continue it for any great length of time for 
want of material. Writing in 1900, Burnham's comment was: 
"Since that time more than one thousand new double stars 
have been added to my own catalogues, and the prospect of 
future discoveries is as promising and encouraging as when 
the first star was found with the six-inch telescope." 

Working with the eighteen and one-half inch refractor of 
the Dearborn Observatory, G. W. Hough discovered 648 
double stars in the quarter-century from 1881 to 1906. In 
1896 and 1897, T. J. J. See, assisted by W. A. Cogshall and 
S. L. Boothroyd, examined the stars in the zone from —20° 
to —45° declination, and in half of the zone (from 4^ to 16^ 
R. A.) from —45° to —65° declination with the twenty-four- 
inch refractor of the Lowell Observatory, and discovered 500 
new double stars. See states that not less than 100,000 stars 
were examined, "many of them, doubtless, on several occa- 
sions." This is probably an overestimate for it leads to a 
remarkably small percentage of discoveries. 

In England, in 1901, the Rev. T. E. H. Espin began pub- 
lishing lists of new double stars discovered with his seventeen 
and one-fourth inch reflector. The first list contained pairs 
casually discovered in the course of other work; later, Mr. 
Espin undertook the systematic observation of all the stars in 



24 THE BINARY STARS 

the Bonn Durchmusterung north of +30°, recording, and, as 
far as possible measuring, all pairs under 10" not already 
known as double. At this writing, his published discoveries 
have reached the total of 1,356. 

In France, M. Robert Jonckheere began double star work 
in 1909 at the Observatoire D'Hem and has discovered 1,319 
new pairs to date. Since 1914, he has been at Greenwich, 
England, and has continued his work with the twenty-eight- 
inch refractor. The majority of his double stars, though close, 
are quite faint, a large percentage of them being fainter than 
the 9.5 magnitude limit of the Bonn Durchmusterung. 

Shorter lists of discoveries have been published by E. S. 
Holden, F. Kiistner, H. A. Howe, O. Stone, Alvan and 
A. G. Clark, E. E. Barnard, and others, and many doubles 
were first noted by the various observers participating in 
the preparation of the great Astronomische Gesellschaft 
Catalogue. 

My own work in this field of astronomy began when I came 
to the Lick Observatory in June, 1895. At first my time was 
devoted to the measurement with the twelve-inch refractor of 
a list of stars selected by Professor Barnard, and the work was 
done under his direction. Later, longer lists were measured 
both with this telescope and with the thirty-six-inch refractor; 
and in selecting the stars for measurement I had the benefit of 
advice — so generously given by him to many double star 
observers of my generation — from Professor Burnham, then at 
the Yerkes Observatory. My attention was early drawn to 
questions relating to double star statistics, and before long the 
conviction was reached that a prerequisite to any satisfactory 
statistical study of double star distribution was a re-survey of 
the sky with a large modern telescope that should be carried 
to a definite limiting magnitude. I decided to undertake such 
a survey, and, adopting the magnitude 9.0 of the Bonn Durch- 
musterung as a limit, began the preparation of charts of con- 
venient size and scale showing every star in the B. D. as bright 
as 9.0 magnitude, with notes to mark those already known to 
be double. The actual work of comparing these charts with 
the sky was begun early in April, 1899. 



THE BINARY STARS 25 

Professor W. J. Hussey, who came to the Lick Observatory 
in January, 1896, also soon took up the observation of double 
stars. His first list consisted of miscellaneous stars, but, in 
1898, he began the remeasurement of all of the double stars 
discovered by Otto Struve, including the 'rejected' pairs. This 
work was carried out with such energy and skill that in 1901, 
in Volume V of the Lick Observatory Publications, a catalogue 
of the OS stars was published which contained not only 
Hussey's measures of every pair but also a complete collection 
of all other published measures of these stars, with references 
to the original publications, and discussions of the motion 
shown by the various systems. In the course of this work, 
Hussey had found an occasional new double star and had 
decided that at its conclusion he would make more thorough 
search for new pairs. In July, 1899, we accordingly combined 
forces for the survey of the entire sky from the North Pole to 
— 22° decHnation on the plan which I had already begun to 
put into execution; Hussey, however, charted also the 9.1 
B.D. stars. Each observer undertook to examine about half 
the sky area, in zones 4° wide in declination. When Mr. 
Hussey left the Lick Observatory in 1905, I took over his zones 
in addition to those assigned to me in our division of the work 
and early in 1915 completed the entire survey to —22° declina- 
tion, as originally planned, between 13^ and i^ in right ascen- 
sion, and to — 14° declination in the remaining twelve hours. 
These come to the meridian in our winter months when condi- 
tions are rarely satisfactory for work at low altitudes. To 
complete the work to —22° in these hours would require several 
years. 

The survey has resulted in the discovery of more than 4,300 
new pairs, 1,329 by Hussey, the others by me, practically all 
of which fall within the distance limit of 5". The statistical 
conclusions which I have drawn from this material will be 
presented in a later chapter. 

It may seem that undue emphasis has been placed upon the 
discovery of double stars in this historical sketch. That a par- 
ticular star is or is not double is indeed of relatively little con- 
sequence; the important thing is to secure accurate measures 



26 



THE BINARY STARS 



through a period of time sufficiently long to provide the data 
for a definite determination of the orbit of the system. But 
the discovery must precede the measures, as Sir John Herschel 
said long ago; moreover, such surveys as that of Struve and 
the one recently completed at the Lick Observatory afford the 
only basis for statistical investigations relating to the number 
and spatial distribution of the double stars. Further, the 
comparison of the distance limits adopted by the successive 
discoverers of double stars and an analysis of the actual dis- 
tances of the pairs in their catalogues affords the most con- 
venient measure of the progress made in the 140 years since 
Herschel began his work, both in the power of the telescopes 
available and in the knowledge of the requirements for advance 
in this field of astronomy. 

The data in the first four lines of the following table are 
taken from Burnham's General Catalogue of his own discov- 
eries, and in the last two lines I have added the corresponding 
figures for the Lick Observatory double star survey, to 191 6. 

The Percentage of Close Pairs in Certain Catalogues of 
Double Stars: 





Class I 
Number 
of Stars 


Class II 
Number 
of Stars 


Sum 


Per- 
centage 
of Close 

Pairs 


William Herschel, Catalogue of 812 Stars 


12 


24 


36 


4-5 


Wilhelm Struve, Catalogue of 2,640 Stars 


91 


314 


405 


150 


Otto Struve, Catalogue of 547 Stars 


154 


63 


217 


40.0 


Burnham, Catalogue of 1,260 Stars 


385 


305 


690 


55.0 


Hussey, Catalogue of 1,327 Stars 


674 


310 


984 


74.2 


Aitken, Catalogue of 2,900 Stars 


1,502 


657 


2,159 


74-4 



The increasing percentage of close pairs is of course due in 
part to the earlier discovery of the wider pairs, but the absolute 
numbers of the closer pairs testify to the increase of telescopic 
power in the period since 1780. If Class I had been divided 
into two sub-classes including pairs under 0.50" and pairs 
between 0.51'' and i.oo", respectively, the figures would have 
been even more eloquent, for sixty per cent, of the Class I pairs 



THE BINARY STARS 



27 



in the last two Catalogues enumerated have measured dis- 
tances of 0.50" or less. 

While the modern period is thus characterized by the num- 
ber of visual binaries, and, in particular, those of very small 
angular distance discovered within it, it is still more notable 
for the development of an entirely new field in binary star 
astronomy. In August, 1889, Professor E. C. Pickering an- 
nounced that certain lines in the objective-prism spectrograms 
of f Ursae Majoris {Mizar) were double on some plates, single 
on others, the cycle being completed in about 104 days.^ An 

Figure I. A, A', A". A'" = 
primary star at points of 
maximum, minimum and 
mean radial velocity. 

B, B', B", B'" = position of 
the companion star at the 
corresponding instants. 

C is the center of gravity of 
the system. There is no star 
or other body at this point. 



explanation of the phenomenon was found in the hypothesis 
that the star consisted of two components, approximately equal 
in brightness, in rapid revolution about their center of mass. 
If the orbit plane of such a system is inclined at a consider- 
able angle to the plane of projection, the velocities in the line 
of sight of the two components will vary periodically, as is 
evident from Figure i ; and, on the Doppler-Fizeau principle,^ 
there will be a slight displacement of the lines of the spectrum 
of each component from their mean positions toward the violet 
end when that component is approaching the Earth, relatively 
to the motion of the center of mass of the system, and toward 
the red end when it is receding, relatively. It is clear from the 
figure that when one component is approaching the Earth, 
relatively, the other will be receding, and that the lines of the 
two spectra at such times will be displaced in opposite direc- 

1 The real period, deduced from many plates taken with slit-spectrographs, is about 
one-fifth of this value, a little more than 20.5 days. 
» Explained in Chapter V. 




28 THE BINARY STARS 

tions, thus appearing double on the spectrograms. Twice, also, 
in each revolution the orbital motion of the two components 
will evidently be directly across the line of sight and the radial 
velocity of each at these times is the same, and is equal to that 
of the system as a whole. The lines of the two spectra, if 
similar, will then coincide and appear single on the plates. 
There is no question but that this explanation is the correct 
one, and Mizar therefore has the honor of being the first star 
discovered to be a spectroscopic binary system. 

A moment's consideration is enough to show that if one of 
the two components in such a system is relatively faint or 
'dark' only one set of spectral lines, that produced by the 
brighter star, will appear upon the plate, but that these will 
be shifted periodically from their mean positions just as are 
the lines in the double spectrum of Mizar. If the plane of the 
system lies so nearly in the line of sight that each star partly 
or completely eclipses the other once in every revolution the 
presence of the darker star may be revealed by a periodic 
dimming of the light of the brighter one; if the orbit plane, as 
will more commonly happen, is inclined at such an angle to the 
line of sight that there is no occultation or eclipse of the stars 
for observers on the Earth the variable radial velocity of the 
brighter star will be the sole evidence of the existence of its 
companion. 

Algol (jS Persei) is a variable star whose light remains nearly 
constant about four-fifths of the time; but once in every two 
and one-half days it rapidly loses brightness and then in a few 
hours' time returns to its normal brilliancy. As early as 1782, 
Goodericke, the discoverer of the phenomenon, advanced the 
theory that the periodic loss of light was due to the partial 
eclipse of the bright star by a (relatively) dark companion. 
In November, 1889, Professor Hermann Vogel, who had been 
photographing the spectrum of the star at Potsdam, announced 
that this theory was correct, for his spectrograms showed that 
before light minimum the spectral lines were shifted toward 
the red from their mean position by an amount corresponding 
to a velocity of recession from the Earth of about twenty-seven 
miles a second. While the star was recovering its brightness. 



THE BINARY STARS 29 

on the other hand, the shift of the lines toward the violet indi- 
cated a somewhat greater velocity of approach, and the period 
of revolution determined by means of the curve plotted from 
the observed radial velocities was identical with the period of 
light variation. Algol thus became the second known spectro- 
scopic binary star and the first of the special class later called 
eclipsing binaries. 

Within a few months two other spectroscopic binary stars 
were discovered; jS Aurigae by Miss Maury at the Harvard 
College Observatory from the doubling of the lines in its spec- 
trum at intervals of slightly less than two days (the complete 
revolution period is 3.96 days), and a Virginis, by Vogel. The 
latter star was not variable in its light, like Algol, nor did its 
spectrum show a periodic doubling of the lines,^ like Mizar and 
/3 Aurigae, but the lines of the single spectrum were displaced 
periodically, proving that the star's radial velocity varied, and 
the cycle of variation was repeated every four days, a Virginis 
is thus the first representative of that class of spectroscopic 
binary systems in which one component is relatively dark, as 
in the case of Algol, but in which the orbit plane does not coin- 
cide even approximately with the line of sight. It is to this 
class that the great majority of spectroscopic binary stars now 
known belong. The reader must not infer that the companion 
stars in systems of this class emit no light; the expression 
relatively dark may simply mean that the companion is two or 
three magnitudes fainter than its primary. If the latter were 
not present, the companion in many systems would be recog- 
nized as a bright star; even the companion of Algol radiates 
enough light to permit the secondary eclipse, when the primary 
star is the occulting body, to be detected by our delicate 
modern photometers. 

The story of the modern spectrograph and its revelations of 
the chemical composition of the stars and nebulae and of the 
physical conditions which prevail in them is a marvelous one, 
but this is not the place to tell it. We must limit ourselves to 
the simple statement that in the years since 1889 the spectro- 
graph has also given us a vast amount of information with 

»The secondary spectrum of a Virginis has been photographed in more recent years. 



30 THE BINARY STARS 

regard to the radial velocities of the stars and, as a by-product, 
with regard to spectroscopic binary systems. In this develop- 
ment the Lick Observatory has taken a leading part, for by the 
application of sound engineering principles in the design of the 
Mills spectrograph, and by patient and skilful experimental 
work extended over several years, Dr. Campbell was enabled, 
in the late 1890's, to secure an accuracy of measurement of 
radial velocity far surpassing any previously attained. The 
New Mills spectrograph, mounted in 1903, led to even better 
results, and it is now possible, in the more favorable cases, to 
detect a variation in the radial velocity even if the range is only 
one and one-half kilometers per second. Other observers and 
institutions have also been most active and successful, and the 
number of known spectroscopic binaries has increased with 
great rapidity. The First Catalogue of Spectroscopic Binaries^ 
compiled by Campbell and Curtis to include the systems ob- 
served to January i, 1905, had 140 entries; by January I, 
1 910, when Dr. Campbell prepared his Second Catalogue of 
Spectroscopic Binary Stars, the number had grown to 306, and 
the Third Catalogue, now in preparation, will contain at least 
596 entries. 

A preliminary count of the last named Catalogue results in 
the following table which gives the distribution of these dis- 
coveries by observatories: 



Lick Observatory, Mount Hamilton 

D. O. Mills Station of the Lick Observatory, at Santiago, Chile 

Yerkes Observatory, at Williams Bay 

Solar Observatory, Mount Wilson 

Other Observatories in the United States and in Canada 

European Observatories 



186 
146 

134 
70 

33 
27 



596 



A slightly different distribution will doubtless result from 
the final count but the table clearly shows that American 
observatories have made this field of research peculiarly their 
own. The Pulkowa, Potsdam, and Bonn observatories are the 



THE BINARY STARS 3I 

three in Europe which are giving most attention to the mea- 
surement of stellar radial velocities. 

The discoveries of the spectroscopic binary stars are here 
credited to observatories rather than individuals because it is 
often a matter for fine discrimination to decide with whom the 
credit for a particular discovery should rest. In general, at 
least three spectrograms are required to prove that a star is a 
spectroscopic binary star. These may all be taken and mea- 
sured by a single observer, or the three plates may be exposed 
by as many different observers in the course of carrying out 
a program of work planned by a fourth; the plates may be 
measured by one or more of the four or by others; variation in 
the radial velocity may be suspected from the second plate and 
confirmed by the third or only by a fourth or still later plate. 
The program for stellar radial velocity determination for the 
Lick Observatory and its auxiliary station, the D. O. Mills 
Observatory, in Chile, for example, is planned, and its execu- 
tion supervised by Dr. Campbell ; Wright, Curtis, Moore, R. E. 
Wilson, Burns, Paddock, and perhaps a score of Fellows and 
Assistants have been associated with him in the actual work. 
In his three catalogues of Spectroscopic Binaries, Campbell 
credits the discovery of the spectroscopic binary stars found in 
the course of this work to the individual who detected the vari- 
ation in radial velocity from his measures of the plates. 

The problem of finding the elements of the orbit of a spec- 
troscopic binary from the data given by the measures of radial 
velocity was solved as early as 1891 by Rambaut, and in 1894, 
Lehmann-Filhes published the method which has been the 
chief one used ever since. A number of other methods have 
been proposed in more recent years, some analytical, others 
graphical, and doubtless others still will be developed. This 
phase of the subject is treated in Chapter VI. 

At the present time orbits for 137 systems have been com- 
puted, a number exceeding that of the visual binary systems 
with known orbits. The reason is not far to seek. The visual 
binaries are systems of vast dimensions and their revolution 
periods range from a minimum (so far as known at present) of 
five and seven-tenths years to a maximum that is certainly 



32 THE BINARY STARS 

greater than 500 years and that may exceed a thousand. 
Castor, for example, was one of the first double stars to be 
observed, and it was the one in which the fact of orbital motion 
was first demonstrated ; but although the observations extend 
from the year 1 719 to date, the length of the revolution period 
is still quite uncertain. The spectroscopic binary stars, on the 
other hand, are, in general, systems of relatively small dimen- 
sions, the revolution periods ranging from five or six hours, as 
a minimum, to a few years. The masses of the systems being 
assumed to be of the same order, the smaller the dimensions, 
the greater the orbital velocity, and the greater the probability 
of the detection of the system by means of the spectrograph, 
for the amount of the displacement of the lines in the spectrum 
is a function of the radial velocity of the star. 

Now if the revolution of a system is accomplished in, say 
two or three days, it is obviously possible for an observer to 
secure ample data for the computation of its definitive orbit 
in a single season. Indeed, if the spectrograph is devoted to 
this purpose exclusively and the percentage of clear nights is 
large, a single telescope may in one season secure the data for 
the orbits of twenty or more systems. 

As in the work of their discovery, so in the computation 
of the orbits of the spectroscopic binary stars the American 
observatories are taking the lead. The Dominion Observatory, 
at Ottawa, Canada, is making a specialty of this phase of the 
work and its observers, notwithstanding the handicap of a none 
too favorable climate for observing work, have more orbits to 
their credit than those of any other institution. The Lick and 
the Allegheny observatories follow quite closely, and the 
Yerkes and Detroit observatories have made valuable contri- 
butions. 

While the spectroscopic binary stars have been receiving ever 
increasing attention in recent years, the visual binary stars are 
by no means being neglected. The work of measuring and 
remeasuring the double stars discovered has been carried on 
enthusiastically by scores of able observers with small tele- 
scopes and with large ones. It is impossible to comment upon 
all of these or to give details of the hundreds of series of mea- 



THE BINARY STARS 33 

sures they have published. But I cannot refrain from referring 
here to two of the most prominent observers of the generation 
that has just passed away — G. V. SchiaparelH and Asaph Hall. 
Schiaparelli's measures are published in two quarto volumes, 
the first containing the measures made at Milan with the eight- 
inch refractor, in the years 1875 to 1885; the second, the series 
made with the eighteen-inch refractor at the same observatory 
in the interval from 1886 to 1900. Hall's work, carried out 
with the twenty-six-inch refractor of the United States Naval 
Observatory at Washington, is also printed in two quarto vol- 
umes, the first containing the measures made in the years 1875 
to 1880; the second, those made from 1880 to 189 1. The work- 
ing lists of both observers were drawn principally from the 
Dorpat and Pulkowa catalogues, but include many of Burn- 
ham's discoveries and some made by Hough and by others. 
The high accuracy of their measures and the fact that they — 
and SchiaparelH in particular — repeated the measures of the 
more interesting stars year after year makes the work of these 
observers of the greatest importance. 

At present, double stars are regularly measured at a num- 
ber of the largest observatories of this country, at several 
important observatories in England and on the continent of 
Europe, and by many enthusiastic amateurs in this country 
and abroad. So voluminous is the literature of the subject 
that one who wishes to trace the full record of one of the dou- 
ble stars discovered by Herschel or by Struve in the original 
sources must have access to a large astronomical library. This 
condition was recognized many years ago, and as early as 
1874 Sir John Herschel's "A Catalogue of 10,300 Multiple and 
Double Stars, Arranged in Order of R. A." was published as a 
Memoir of the Royal Astronomical Society. This catalogue 
attempted merely to give a consecutive list of the known 
double stars, without measures, and did not go far towards 
meeting the needs of observers or computers. The first really 
serviceable compendium was that published by Flammarion 
in 1878, entitled "Catalogue des Etoiles Doubles et Multiples 
en Mouvement relatif certain." The volume aimed to include 
all pairs known from the actual measures to be in motion; 819 



34 THE BINARY STARS 

systems are listed, each with a fairly complete collection of 
the published measures, about 14,000 in all, and notes on the 
nature of the motion. For thirty years this book formed a 
most excellent guide to observers. 

The following year, 1879, "A Handbook of Double Stars," 
prepared by Crossley, Gledhill, and Wilson, was published in 
London — a work that has had a wide circulation and that has 
proved of the greatest service to students of double star 
astronomy. It is divided into three parts, the first two giving 
a general account of double star discoveries and methods of 
observing and of orbit computation. The third section con- 
tains a "Catalogue of 1,200 double stars and extensive lists of 
measures." An appendix gives a list of the principal papers 
on double stars. 

* In 1900, Burnham published a General Catalogue of his 
own discoveries containing a complete collection of all known 
measures of these stars with notes discussing the motion when 
such was apparent, and references to the original sources 
from which the measures were taken. This proved to be the 
first of a series of such volumes. Hussey's catalogue of the 
Otto Struve stars, to which reference has already been made, 
was published in 1901, and five years later, in 1906, Lewis's 
great volume on the Struve stars appeared. This is, in effect, 
a revision of the Mensiirae Micrometricae and gives all of the 
2 stars in the order of their original numbers, disregarding the 
inconvenient division into classes. Such of the S 'rejected' 
stars as have been measured by later observers are also in- 
cluded, and all or nearly all of the published measures of each 
pair. The notes give an analysis and discussion of the motions 
which have been observed, and form one of the most valuable 
features of the work, for the author has devoted many years 
to a comprehensive study of double star astronomy in all its 
phases. In 1907, Eric Doolittle published a catalogue of the 
Hough stars, all of which he had himself reobserved, and in 
1 91 5, Fox included in Volume I of The Annals of the Dearborn 
Observatory catalogues of the discoveries of Holden and of 
Kiistner with a new series of measures of these stars. Thus all 
of the longer catalogues of new double stars, except the very 



THE BINARY STARS 35 

recent ones and those of Sir John Herschel, have now been 
revised and brought up to date, for Sir WilHam Herschel's 
discoveries, except the very wide pairs, are practically all in- 
cluded in the Mensurae Micrometricae. 

Every one of the volumes named is most convenient for 
reference and each one contains information not easily to be 
found elsewhere; but they are all surpassed by Burnham's 
comprehensive and indispensable work, A General Catalogue of 
Double Stars within 121° of the North Pole, which was published 
by The Carnegie Institution of Washington in 1906. This 
monumental work consists of two parts, printed in separate 
quarto volumes. Part I contains a catalogue of 13,665 double 
stars, including essentially every pair, close or wide, within 
the sky area named, that had been listed as a double star 
before 1906. The positions, for 1880, are given, with the dis- 
covery date and measure or estimate. Part II contains 
measures, notes and complete references to all published papers 
relating to each pair. This great work in itself is an ample 
guide to anyone who wishes to undertake the measurement of 
double stars and desires to give his attention to those pairs, 
not very recent discoveries, which are most in need of obser- 
vation. When Burnham retired from active astronomical 
work, he turned over to Professor Eric Doolittle, of the Flower 
Observatory, all material he had accumulated since the ap- 
pearance of the General Catalogue; and Doolittle has since 
kept a complete record of every published measure and orbit, 
with the view of printing an extension to the catalogue when 
the need for it is manifest.^ 

It has been convenient, in this narrative, to confine atten- 
tion until now to the double star work done at observatories 
in the northern hemisphere, for it has been there that this 

* M. Robert Jonckheere has just published, in the Memoirs of the Royal Astronomical 
Society (vol. Ixi, 1917), a Catalogue and Measures of Double Stars discovered visually from 
1005 to IQ16 within 105° of the North Pole and under 5' Separation. This is, in effect, an 
extension of Burnham's General Catalogue, though the author has excluded pairs wider than 
5 ' instead of recording every pair announced by its discoverer as double and has adopted a 
more northern sky limit than Burnham's. The volume is particularly valuable because it 
gives in collected form Jonckheere's own discoveries with measures at a second epoch as 
well as at the time of discovery. The other long lists in the volume are Espin's discoveries 
and those made at the Lick Observatory; in all, there are 3.9SO entries, sufficient evidence 
of the activity of double star discoverers in recent years. 



36 THE BINARY STARS 

branch of astronomy has received most attention. Even today 
there are relatively few telescopes in the southern hemisphere 
and only two or three of these are in use in the observation of 
double stars. But the state of our knowledge of the southern 
double stars is better than this fact would indicate. Many 
stars south of the equator have been discovered from stations 
in the northern hemisphere, and the few southern workers in 
this field have made a most honorable record. 

We have seen that two of the earliest double stars discov- 
ered — a Centauri and a Cruets — were stars not visible from 
European latitudes; but the first extensive list of double stars 
collected at a southern observatory was James Dunlop's cata- 
logue of 253 pairs observed at Parametta, N. S. W., in the 
years 1825-182 7 with a nine-foot reflecting telescope. These 
stars, however, are as a rule very wide pairs and are of com- 
paratively little interest. A few double stars are contained in 
Brisbane's Parametta catalogue, published in 1835, and more 
in the later meridian catalogues of the Royal Observatory at 
the Cape of Good Hope, the Argentine National Observatory 
at Cordoba, and of other southern observatories. 

The most important early paper on southern double stars 
is beyond question the chapter upon them in Sir John Her- 
schel's Results of Astronomical Observations made during the 
Years 1834, 1835, 1836, 1837 , 1838 at the Cape of Good Hope 
which was published in 1847. Innes says, "The sections on 
double stars in this work are to the southern heavens what 
Struve's Mensurae Micrometricae are to the northern heavens." 
A catalogue of the discoveries made at Feldhausen, C. G. H., 
with the twenty-foot reflector is given, which contains the 
pairs h3347 to h5449, together with measures of such previ- 
ously known pairs as were encountered in the 'sweeps*. Many 
of the new pairs are wide and faint, resembling the h stars 
discovered at Slough, in England; but many others are com- 
paratively close, many are very bright, and a number are 
among the finest double stars in the southern sky. Another 
division of this chapter gives the micrometer measures, made 
with the five-inch refractor, of many of these new pairs and 
of some of the known ones. Innes says that "the angles of the 



THE BINARY STARS 37 

pairs are all through of high excellence"; but Herschel himself 
points out the sources of weakness in his measures of distances. 

Herschel's station at Feldhausen was not a permanent ob- 
servatory, and when he returned to England work there was 
discontinued; nor was double star work seriously pursued at 
any other southern station until about forty years later. In 
1882, a list of 350 new pairs was published by H. C. Russell, 
director of the Sydney Observatory, N. S. W., the measures 
being made by Russell and by L. Hargrave. In 1884, an 
additional list of 130 pairs, mostly wide, was published, and 
in the following years several lists of measures by these 
observers and their colleague, J. A. Pollock, a few of the 
measured pairs being new. In 1893, R. P. Sellors published a 
short list (fourteen pairs, all under 2") discovered by him at 
the same observatory, and in the following years he contri- 
buted many measures of known pairs and discoveries of a 
few additional new ones. 

The man of the present generation who has done most to 
advance double star astronomy in the southern hemisphere is 
R. T. A. Innes, now Government Astronomer at the Union 
Observatory, Johannesburg, Union of South Africa. In 1895, 
he published a list of twenty-six pairs 'probably new' which 
were found with a six and one-fourth-inch refractor at Sydney, 
N. S. W., and the following year, sixteen additional discoveries 
made with a small reflector. In this year, 1896, Mr. Innes 
joined the staff of the Royal Observatory of the Cape of 
Good Hope and there, in addition to his regular duties, con- 
tinued his double star work with the seven-inch refractor, and 
later, for a time, with the eighteen-inch McClean refractor. 
With these instruments he brought the total of his discoveries 
to 432 and made a fine series of measures. Since going to his 
present station in 1903, he has discovered more than 600 
additional pairs with a nine-inch refractor and has made 
extensive series of measures which are of the greatest impor- 
tance not alone because of the stars measured but also be- 
cause the work has been most carefully planned to eliminate 
systematic errors of measure as far as possible. A large 
modern refractor is to be erected at the Union Observatory as 



38 THE BINARY STARS 

soon as the glass disks can be secured, and it is Mr. Innes's 
intention to use this instrument in an even more systematic 
study of double stars. 

Another telescope was set to work upon southern double 
stars when Professor Hussey accepted the directorship of the 
observatory of the La Plata University, Argentina, in 191 1, in 
addition to his duties at Ann Arbor, Michigan. Mr. Hussey 
has spent several periods at La Plata organizing the work of 
the observatory and personally using the seventeen-inch re- 
fractor in searching for, and measuring double stars. So far 
the work has resulted in the publication of two lists of new 
pairs containing 312 stars and of a valuable list of measures 
of known pairs. 

Mr. Innes has shown that even before he began his own 
work the number of close double stars known in the sky area 
south of —19° declination exceeded the number in correspond- 
ing distance classes north of +19° declination contained in 
the Mensurae Micrometricae. His own discoveries and those 
by Hussey at La Plata consist almost entirely of close pairs, 
and we may allow his claim, without serious protest, that in 
point of double star discovery the southern hemisphere is not 
greatly in arrears. If Innes and Hussey are able to carry out 
their programs for systematic surveys of the sky to the South 
Pole, we shall have as complete data for statistical studies of 
the southern double stars as we now have for those of the 
northern pairs. 

In 1899, Innes published his Reference Catalogue of Southern 
Double. Stars which has proved to be a most valuable work. 
The object was to include "all known double stars having 
southern declination at the equinox of 1900"; but the author 
did not follow the plan adopted by Burnham of including all 
objects published as double stars regardless of angular separa- 
tion. Instead, he adopted limits which varied with the mag- 
nitudes of the stars, ranging from i " for stars of the ninth mag- 
nitude to 30" for those of the first magnitude. This course 
has been sharply criticized by some writers, but I think there 
can be no serious question as to the .soundness of the principle 
involved. Whether the limits actually adopted are those best 



THE BINARY STARS 39 

calculated to promote the progress of double star astronomy 
is a different matter and raises a question to which more atten- 
tion will be given on a later page. Mr. Innes has in prepara- 
tion a new edition of the Reference Catalogue, bringing it up 
to the present date. Possibly he may adopt in it some modi- 
fications of his former limits. A feature of the work to which 
reference must be made is the excellent bibliography of double 
star literature which forms the appendix. 

Our knowledge of the spectroscopic binary stars in the far 
southern skies is due almost entirely to the work carried on 
at the D. O. Mills Station of the Lick Observatory, established 
at Santiago, Chile, in 1893. The instrumental equipment 
consists of a thirty-seven and one-fourth-inch silver-on-glass 
reflector and spectrographs similar in design to those in use 
on Mount Hamilton. The working program is the measure- 
ment of the radial velocities of the stars and nebulae which are 
too far south to be photographed at the Lick Observatory 
itself. The discovery of binary stars is not the object in view, 
but the table given on page 30 shows that more than one-fifth 
of the entire number of these systems known at the present 
time have been found at this Station in the fourteen years of 
its existence. When we add to this number the spectroscopic 
binary stars with southern declinations which have been 
detected by observers at stations in the northern hemisphere, 
we shall find that in this field there is no disparity whatever 
between the two hemispheres of the sky. 



CHAPTER III 

OBSERVING METHODS, VISUAL BINARY STARS 

The operation of measuring a double star is a very simple 
one. The object is to define at a given instant the position of 
one star, called the companion, with respect to the other, 
known as the primary. When the two stars are of unequal 
magnitude the brighter is chosen as the primary; when they 
are of equal brightness, it is customary to accept the dis- 
coverer's designations. 

From the first work by Sir William Herschel, the measures 
have been made in polar coordinates; and since about 1828, 
when Sir John Herschel recommended the practice, the posi- 
tion angle has been referred to the North Pole as zero point 
and has been counted through 360°. 

That is, the position angle is the angle at the primary star 
between the line drawn from it to the North Pole and one 
drawn from it to the companion, the angle increasing from 
zero when the companion is directly north through 90° when 
it is at the east, 180° when it is south, 270° when it is west, 
up to 360° when it is once more directly north. The dista^ice 
is the angular separation between the two stars measured at 
right angles to the line joining their centers. The two co- 
ordinates are usually designated by the Greek letters 6 and p, 
or by the English letters p and s. 

THE MICROMETER 

Experience has proved that the parallel-wire micrometer is 
the best instrument for double star measurements. A com- 
plete description of it is not necessary here. For this the reader 
is referred to Gill's article on the Micrometer in the Encyclo- 
pedia Britannica. Essentially it consists of a tube or adapter 
firmly fitted into the eye-end of the telescope and carrying on 
its outer end a graduated circle (the position circle) reading 



THE BINARY STARS 4I 

from 0° to 360° in a direction contrary to the figures on a 
clock dial. A circular plate fitting closely within the position 
circle and adjusted to turn freely within it carries an index, or 
a vernier, or both, to give the circle reading. In the microme- 
ters in use at the Lick Observatory, this plate is rotated about 
the optical axis of the telescope by an arm carrying a pinion 
which meshes into rack teeth cut on the outer circumference 
of the position circle. A clamp is provided to hold the plate 
and circle together at any desired reading, and a tangent screw 
to give a slow motion. Upon the vernier plate an oblong box 
is mounted within which the parallel wires or threads (they 
are usually spider lines) are placed. This box is movable 
longitudinally by a well-cut, but not very fine screw. One 
thread, the fixed thread, is attached to the inner side of the 
upper plate of the box, and the other, the micrometer or mov- 
able thread, is attached to a frame or fork which slides freely 
in the box longitudinally, but without any lateral play. The 
fork is moved by a very fine and accurately cut screw which 
enters the box at one end. At its outer extremity, this screw 
carries a milled head divided into 100 parts, the readings in- 
creasing as the screw draws the micrometer thread towards 
the head. Strong springs at the opposite end of the fork 
carrying this thread prevent slack, or lost motion. 

The two threads, the fixed and the micrometer, must be so 
nearly in the same plane — the focal plane of the objective — 
that they can be brought into sharp focus simultaneously in an 
eye-piece of any power that may be used, but at the same time 
must pass each other freely, without the slightest interference. 
Instead of a single fixed thread, some micrometers carry sys- 
tems of two, three, or more fixed threads, and frequently 
also one or more fixed transverse threads. Some also sub- 
stitute two parallel threads separated a few seconds of arc for 
the Single movable thread. For double star work, the simple 
micrometer with only two threads is unquestionably to be 
preferred, and even for comet, asteroid, satellite, and other 
forms of micrometric work, I regard it as superior to the more 
complicated forms and less liable to lead to mistakes of 
record. 



42 THE BINARY STARS 

The telescope is assumed to be mounted stably and to be in 
good adjustment. Assured as to these two points and as to the 
firm attachment of the micrometer to the telescope tube that 
the zero reading of his position circle shall remain constant, 
the double star observer has still to determine the value of one 
revolution of his micrometer screw and the zero or north point 
reading of his position circle before beginning actual measure- 
ment. The reading for coincidence of the threads is elimi- 
nated by the method of double distance measures, as will be 
shown presently, and the distances themselves are, in general, 
so small, and modern screws so accurate, that irregularities 
in the screw and corrections for temperature may be regarded 
as negligible. If desired, however, they may be determined 
in connection with measures for the revolution value. 

THE ZERO POINT 

The determination of the zero point will be considered first. 
The simplest practical method, and the one adopted by ob- 
servers generally, is to put on the lowest power eye-piece which 
utilizes the entire beam of light, direct the telescope upon an 
equatorial star near the meridian, stop the driving-clock, and 
turn the micrometer by the box screw and the position-circle 
pinion until the star 'trails' along the thread across the entire 
field of view. The star should be bright enough to be seen 
easily behind the thread, but not too bright. With the twelve- 
inch telescope I find a star of the seventh or eighth magnitude 
most satisfactory; with the thirty-six-inch, one of the ninth 
or tenth magnitude. A little practice will enable the observer 
to determine his 'parallel' reading with an uncertainty not 
greater than one-fifth of one division of his circle. On the 
micrometer used with the thirty-six-inch telescope, this 
amounts to 0.05°. Several independent determinations should 
be made. If the micrometer is not removed from the tele^ope 
and is set firmly to the tube, it is probable that the parallel 
reading need be checked only once or twice a week. When, 
as at the Lick Observatory, the micrometer is liable to be 
removed almost any day and is certainly removed several 
times every week, the observer very promptly forms the 



THE BINARY STARS 43 

habit of determining the parallel at the beginning of his work 
eoery night; my own practice is to check the value at the close 
of work also. 

90° added to the parallel gives the north point or zero reading. 

REVOLUTION OF THE MICROMETER SCREW 

The value of the revolution of the micrometer screw should 
be determined with the greatest care and the investigation 
should be repeated after a reasonable time interval to detect 
any wear of the screw. Two different methods of procedure 
are about equally favored by observers: the method of trans- 
its of circumpolar stars and the method of direct measures of 
the difference in declination of suitable pairs of stars. 

In the first method the position circle is set for the zero 
reading {i. e,, 90° from the reading for parallel) and the tele- 
scope turned upon the star a short time before it culminates. 
(The driving clock, of course, is stopped.) Set the micrometer 
thread just in advance of the star as it enters the field of view 
(it is convenient to start with the milled head set at zero of a 
revolution) and note the time of the star's transit either on 
the chronograph or by the eye-and-ear method. Advance the 
thread one revolution or a suitable fraction of a revolution and 
take another transit, and repeat this procedure until the star has 
crossed the entire field of view. A low-power eye-piece should 
be used and the series of measures so planned that they will 
extend over from forty to eighty revolutions of the screw, 
about half of the transits being taken before the star crosses 
the meridian, the other half after. Great care must be taken 
not to disturb the instrument during the course of the obser- 
vations for the slightest changes in its position will introduce 
errors into the measures. It is well to repeat the observations 
on a num.ber of nights, setting the telescope alternately east 
and west of the pier. A sidereal time piece should be used in 
recording the times of transits and if it has a large rate, it 
may be necessary to take this into account. Theoretically, a 
correction for refraction should also be introduced, but if all 
of the measures are made near the meridian, this correction 
will rarely be appreciable. 




44 THE BINARY STARS 

In the figure, let P be the pole, EP the observer's meridian, 
ab the diurnal path of a star, ^5 the position of the micrometer 
thread when at the center of the field p 

and parallel to an hour circle PM, and 4 

BS' any other position of the thread. 
Now let ruo be the micrometer reading, 
to the hour angle, and To the sidereal „., 

time when the star is at 5, and m, /, / / 

and T the corresponding quantities / / 

when the star is at S\ and let R be the 
value of one revolution of the screw. 

Through 5' pass an arc of a great circle 
S'C perpendicular to AS. Then, in the 
triangle CS'P, right-angled at C, we have Figure 2 

CS' = (w - mo)R, S'P = 90° - 5, CPS' = t - to = T - To\ 
and we can write 

sin {{m — mo)R] = sin {T — To) cos 5; (i) 

or, since {m — mo)R is always small, 

(w — nto) R = sin(r — To) cos 5 /sin i". (2) 
Similarly, for another observation, 

(w' -mo)R = sin(r - To) cos 5 / sin i \ (3) 
Combining these to eliminate the zero point, 
(m' - m)R = sin(r - To) cos 5 /sin 1" - sin{T - ro)cos 5/sin i" (4) 

from which the value of R is obtained. The micrometer read- 
ings are supposed to increase with the time.^ 

If eighty transits have been taken, it will be most convenient 
to combine the first and the forty-first, the second and the 
forty-second, and so on, and thus set up forty equations of 
condition of the form of equation (4). The solution of these 
equations by the method of least squares will give the most 
probable value for R. 

If the value of R is to be determined by direct measures of 
the difference of declination between two stars, the stars 
should satisfy the following conditions: they should lie on, 

1 From Campbell's Practical Astronomy. 



THE BINARY STARS 45 

or very nearly on, the same hour circle; their proper motions 
as well as their absolute positions at a given epoch should be 
accurately known; they should be nearly of the same magni- 
tude and, if possible, of nearly the same color; the difference 
of declination should amount to from fifty to one hundred 
revolutions of the micrometer screw; and, since this will 
ordinarily exceed the diameter of the field of view of the eye- 
piece, one or more intermediate stars (whose positions do not 
need to be so accurately known) should lie nearly on the line 
joining them and at convenient intervals to serve as steps. 

There are not many pairs of stars which answer all of the 
requirements. Probably the most available ones are to be 
found in the Pleiades and other open clusters which have 
been triangulated by heliometer observations. 

The measures should be made only on the most favorable 
nights and at times when the stars are high enough in the sky 
to make the correction for refraction small. The difference 
of declination should be measured from north star to south 
star and also in the opposite direction and the measures should 
be repeated on several nights. If extreme accuracy is desired 
in the refraction corrections the thermometer and barometer 
should be read at the beginning and also at the end of each 
set of measures, and if the effect of temperature is to be in- 
cluded in the determination of R, measures must be made at 
as wide a range of temperature as is practicable. 

In making the reductions, the star-places are first brought 
forward from the catalogue epoch to the date of the actual 
observations by correcting rigorously for precession, proper 
motion, and the reduction from mean to apparent place. The 
apparent place of each star must then be corrected for refrac- 
tion. It will generally be sufficiently accurate to use Com- 
stock's formula, in the following form: 

Refraction m 5 = ; — ; — tan z cos g, 

460 + 1 

where z is the apparent zenith distance, and q the parallactic 
angle of the star, b the barometer reading in inches and / the 
temperature of the atmosphere in degrees Fahrenheit. In 



46 



THE BINARY STARS 



practice I have found it more convenient to correct each star 
for refraction in the manner described than to correct the dif- 
ference in decHnation by the use of differential formulae. 

The following pairs of stars in the Pleiades have actually 
been used by Professor Barnard in determining the value of 
one revolution of the micrometer screw of the forty-inch 
telescope of the Yerkes Observatory: 



BD. Mag. BD. Ma, 


I. A5 


+ 23°537 (7-5) and +23^542 (8.2) 696.19 


+ 23°5i6 (4 


8) and H-23°5i3 (9 


0) 285 


94 


+ 23°557 (4 


0) and +23°559 (8 


4) 599 


58 


+ 23°56i (7 


5) and -f 23°562 (7 


8) 479 


II 


+ 23°558 (6 


2) and +23°562 (7 


8) 401 


10 


+ 23°563 (7 


2) and H-23°569 (7 


5) 494 


14 


+ 23*^557 (4 


0) and +23°558 (6 


2) 300 


25 


+ 23°507 (4 


7) and +23^505 (6 


5) 633 


40 



The differences in declination given in the final column are 
for the epoch 1903.0 and are the results of Dr. Elkin's mea- 
sures with the Yale heliometer. Dr. Triimpler has recently 
been making an accurate survey of the Pleiades cluster on the 
basis of photographs taken at the Allegheny Observatory, and 
when his results are published they may, to advantage, be 
combined with the values given above. 

The last pair in the list consists of the bright stars Electra 
and Celaeno, and Professor Barnard kindly permits me to 
print in full his measures of them, made in 1912, to illustrate 
the use of step stars. The step stars in this case are respectively 
of magnitude 11 .0 and 11 .5 and lie nearly, but not quite, on 
the line joining the bright stars. Both the tube of the forty- 
inch telescope and the screw of the micrometer are of steel and 
therefore mutually correct each other in temperature changes, 
at least approximately; but the focal length of the object 
glass is three-fourths of an inch shorter in winter than in 
summer whereas the tube shortens only one-half of an inch. 
A slight correction is therefore necessary if all of the measures 
are to be reduced to the focus for a common temperature. 
The column Scale reading, in the table of measures, gives the 



THE BINARY STARS 



47 





O HN O »0 t^ 00 »o 

O O vO ON O t^ Tj- 


<s 


t^ 


o 


vO lO lO lO lO lO »o 


to 


3 


lO lO lO lO »0 lO »o 


to 


H 


O vO VO vD VO vO VO 


VO 


c 


r^ t^ \0 rO VO O O 


rO 


11 


8 8 8 8 8 8 8 

''o 660606 


8 

d 


o 


1 1 1 1 1 


t 


u 






^ c 


VO 00 -^ 00 11 




11 


<N fC (s cs n cs n 

(S (S M (VJ M (S M 




o g 


Ov <N 







CN n C< CS 1-1 CS (S 


(V» 


o tJ 


0000000 





2 S 


"^6 6 6 6 6 6 6 





2 o 


+++++++ 


+ 


f^^^J. 








B 






10 




< 






X 


(N (N (N « 1- M M 




+ + + + + + + 




-s 


T^ 00 vo 00 Tf 00 »o 


VO 




00 Th Ti- r^ »o 10 rj 


10 


S '« 


10 10 »o 10 to 10 10 


to 


^<1 


^o »o »o »o 10 »o >o 


to 


VO VO vO VO vO VO vO 


VO 


11 


<v, "-1 vO vO vO to 


00 


to to rO ■^ M rO n 


'n 


t^ t>. t^ t^ t^ t^ t^ 


t^ 








* u 


(N (V) (S n (S (S M 


r< 


* 


VO M vO ^ vO vO 


VO 




r» ON 





s 


VO vO 10 vO VO VO VO 


^ 


OJ 


10 10 to to to to to 


to 


c< 


(S (N <N (V) Cv« n M 




* 






13 






Cj 


00 to a^ (V| Tj- \0 Tl- 


rr> 




OV HH fO (S HH ON 




2 <^ 


<S •-> <S C( CI C< HH 


n 


i3 * 








00 00 00 00 00 00 CO 


00 


0) 




ft 








W 








HH n On ►I VO CS ON 




N 


(s fs rt <s c< 




o\ 


i 1 







Tf- 


^ 


^ 


<v 


CO 




rO 


^ 


VO 



0) 

J3 



15 



•5 









(fi 





<u 





s 


10 


en 


0) 


(Tt 


i-i 


dJ 


3 


B 






Ui 


tn 


<l> 


"H 


a 


ct3 


flj 


C 


■M 


u 




0^ 


4-> 


PQ 


cd 



en vO 



48 THE BINARY STARS 

readings for focus on the draw-tube of the telescope and the 
following column, the corrections required to reduce the 
measures to the focal length corresponding to a temperature 
of 50°F. The column H. A . gives the hour-angle at which the 
measures were made. The remaining columns are self- 
explanatory. 

MEASURING A DOUBLE STAR 

When the telescope has been directed upon the star and 
clamped, the star is brought up to the threads by means of the 
screw moving the entire micrometer box. The position angle 
is then measured, and in doing this my practice is to run the 
micrometer thread well to one side of the field of view, bring 
the double star up to the fixed thread by means of the screw 
moving the box and then rotate the micrometer by means of 
the pinion provided, keeping, meanwhile, the fixed thread 
upon the primary star, until the thread also passes centrally 
through the companion star. It is most convenient to manipu- 
late the box screw with the left hand and the pinion with the 
right. 

The tangent screw giving a slow motion in position angle 
is never used; in fact, it has been removed from the 
micrometer. When the seeing is good, the star images round, 
small, and steady, it is easy to hold both images on the thread 
until the eye is assured of their precise bisection. Under less 
favorable conditions a rapid to and fro motion of the box 
screw places the stars alternately on either side of the thread 
while the pinion is being rotated backward and forward until 
the eye is satisfied of the parallelism of the thread to the line 
joining the centers of the star images. 

Ordinarily four independent settings for position angle are 
made, the circle being read, not by the vernier but by an index, 
directly to half degrees in the case of the twelve inch microme- 
ter, to quarter degrees in the case of the thirty-six inch, and 
by estimation to the one-fifth of a division, i. e., to o.i° and 
0.05°, respectively. After each setting the micrometer is 
rotated freely backward and forward, simply by turning the 
box directly with the hands, through an arc of 80° to 100°, 
the eye being removed from the eye-piece. 



THE BINARY STARS 49 

The circle is next set to a reading 90° greater (or less) than 
the mean of the readings for position angle and the distance 
is measured by bisecting one star with the fixed thread, the 
other with the micrometer thread. It is most convenient to 
turn the micrometer screw with the right hand, the box 
screw with the left. Then interchange the threads, placing the 
micrometer thread on the first star, the fixed thread on the 
other. The difference between the two readings of the mi- 
crometer screw-head gives the double-distance, i. e., twice the 
angular separation, and eliminates the zero or coincidence 
reading. Three measures of the double distance are generally 
made. The milled head of the screw, which is divided to 
hundredths of a revolution, is read to the i /looo of a revolu- 
tion by estimation. Care is always taken to run the microme- 
ter thread back several seconds of arc after each setting and 
to make the final turn of the screw at each bisection forward, 
or against the springs.^ 

Any ordinary note book will answer as a record book. At 
the Lick Observatory, we have found convenient a book 7" x 
SK'' containing 150 pages of horizontally ruled, sized paper 
suitable for ink as well as pencil marks. The observing record 
is made in pencil, the reductions with ink. No printed forms 
are necessary or even convenient. A sample entry taken from 
my observing book shows the form of record adopted, and also 
the very simple reductions: 

36* Sat. Jan. 27, 1917. 



So-: 

128.75° 
129.70 . 
129.30 


rauri= 

N. F. 


= 2554 
0.9" ± 

3 


49.401 
.400 
.403 


.581 
.578 
.580 

.580 
.401 

).i79 


Parallel= 10.25° 

4^3 
1000 

2 to 2-1- 

Well separated with 
520- power 


130.40 


49.401 


129.54 
100.25 







29.3''= ^o .089 R= 0.88"= po 

2 The bisection of the star by the fixed thread should be made anew at each setting with 
the micrometer screw, because, under even the best conditions, it cannot be assumed that 
the star images will remain motionless during the time of observation. 



50 THE BINARY STARS 

Two such entries are ordinarily made to the page. The 
column at the left records the four settings for position angle; 
the mean is taken and the reading of the circle for parallel 
plus 90° is subtracted to obtain the position angle. Whether 
this value is the correct one or whether 180° is to be added to 
it is decided by the note made of the quadrant while observing 
— N. F. in the present case. When recording the quadrant, 
which is done after the position angle settings have been 
entered, I record also an estimate of the distance and of the 
difference of magnitude of the components. Sometimes, when 
the companion is very faint, I record, instead, a direct estimate 
of its magnitude. At this time, too, I record, at the right, 
the date, the sidereal time to the tenth of an hour, the power 
of the eye-piece used, an estimate of the seeing on a scale on 
which 5 stands for perfect conditions and any observing notes. 
Measures of distance are then made and recorded. Here the 
reduction consists in taking half the difference of the two 
means and multiplying the result by the value of one revolu- 
tion of the micrometer screw (in this instance 9.907"). 

The results are transfered to a "ledger", the date being re- 
corded as a decimal of the year. The ledger entry for the above 
observation is: 

80 Tauri = S554. 

1917.075 29.3° 0.88" AM = 3 4*\3 1000 2to2-f- bk. 87,147 

the last item being the number and page of the observing book. 
Practically all observers agree in the method of measuring 
the angular distance, but many prefer a somewhat different 
procedure for determining the position angle. They bring the 
two threads fairly close together — to a separation twice or 
three times the diameter of the primary's apparent disk — 
and then, placing the two stars between the threads, turn the 
micrometer until the line joining the stars appears to be parallel 
to the threads. I have found that I can secure equally satis- 
factory measures by this method when the two stars are well 
separated and of nearly equal magnitude, but not when the 
angular distance is small or when the stars differ much in 
brightness. While it may be a matter of personal adaptation 
I incline to think that measures made in this manner are more 



THE BINARY STARS 5I 

likely to be affected by systematic errors than those made by 
the method first described. 

Whatever method is adopted it is of the first importance 
that the head of the observer be so held that the line between 
his eyes is parallel or perpendicular to the line joining the two 
stars. I can make the bisections with more assurance when 
the line between the eyes is parallel to the one joining the two 
stars, and hold my head accordingly unless the line is inclined 
more than 45° to the horizon. Some observers prefer the 
perpendicular position. 

There are some other points that must be taken into con- 
sideration to secure satisfactory results. The star images as 
well as the threads must be brought sharply into focus; the 
images must be symmetrically placed with respect to the 
optical axis; and the threads must be uniformly illuminated 
on either side. In modern micrometers the illumination is 
usually provided by a small incandescent lamp placed in such 
a position that a small mirror can throw the light through a 
narrow opening in one end of the micrometer box. This 
mirror can be rotated through 90° thus permitting a variation 
in the intensity of the light from full illumination to zero. 
Suitable reflectors placed within the micrometer box, at the 
opposite end, insure equality in the illumination on both sides 
of the threads. Glass slides can also be placed in front of the 
opening admitting the light in order to vary its intensity or its 
color as may be desired. I have found no advantage in using 
these. The earlier double star observers frequently illuminated 
the field of view instead of the threads and an occasional ob- 
server still advocates this practice, but the great majority, I 
think, are agreed that this is a less satisfactory arrangement. 

It is hardly necessary to say that the micrometer threads 
must be stretched to a tension sufficient to keep them perfectly 
straight even when the atmosphere is very moist and that they 
must be free from dust or other irregularities and accurately 
parallel. A cocoon of spider thread should be obtained from 
an instrument maker and kept on hand with the necessary 
adjusting tools and the micrometer threads replaced as often 
as they become unsatisfactory. A little practice will enable 



52 THE BINARY STARS 

the observer to set a thread in position in a very short space 
of time; in fact, from Burnham's days to the present time, a 
new thread has frequently been set into the thirty-six-inch 
micrometer during the night and observing been resumed 
within an hour. 

The most important precaution to be taken in double star 
observing is quite independent of instrumental adjustments. 
It is to make measures only on nights when the observing condi- 
tions are good. Measures made under poor observing condi- 
tions are at best of little value, and at worst are a positive 
hindrance to the student of double star motions. They annoy 
or mislead him in his preliminary investigations and are prac- 
tically rejected in his later work. I make this statement with 
all possible emphasis. 

It is of almost equal consequence to select stars suited to 
the power of the telescope employed. This, however, is to a 
considerable extent a matter involving the personal equation. 
A Dawes, a Dembowski, or a Burnham can measure with 
comparatively small apertures stars that other observers find 
difficult with much larger telescopes. 

MAGNITUDE ESTIMATES 

It is well known that the magnitudes assigned to the com- 
ponents of the same double star by different observers fre- 
quently show a range that is excessively large. Whatever excuse 
there may have been for this in earlier days, there is certainly 
little at the present time when the magnitudes of all of the 
brighter stars are given in the photometric catalogues and 
those of all stars to at least 9.5 magnitude in the various 
Durchmusterungs. It is certainly advisable to take the com- 
bined magnitude of the two components (or the magnitude of 
the brighter star, if the companion is very faint) from these 
sources instead of making entirely independent estimates. 
The difference of magnitude is then the only quantity the 
double star observer need estimate. If this difference is not 
too great it can be estimated with comparative accuracy; if 
one component is very faint, a direct estimate of its brightness 
may be based upon the limiting magnitude visible in the tele- 



THE BINARY STARS 53 

scope used, care being taken to allow for the effect of the bright 
companion which will always make the faint star appear 
fainter than it really is. 

To derive the brightness of each component when the com- 
bined magnitude and the difference of magnitude are known, 
we have the relations, A = C + :x;, B = A + ^, in which A and 
B are the magnitudes of the brighter and fainter component, 
respectively, C the combined magnitude, and d the estimated 
difference of magnitude, while x is given by the equation 

log (i + -^'j 

\ 2512 / 

X = 

0.4 

We may tabulate x for different values of d as in the follow- 
ing table which is abbreviated from the one in Innes's Refer- 
ence Catalogue: 



d 


X 


d 


X 


0.0 


0.75 


1-5 


0.25 


0.25 


0.6 


2.0 


0.15 


0.5 


0.5 


2.5 


O.I 


0.75 


0.4 


30 


0.05 


I.O 


0.3 


4.0 


0.0 



To illustrate the use of the table let d, the observed difference 
in brightness, be three-fourths magnitude (it is sufficiently 
accurate to estimate the difference to the nearest quarter 
magnitude), and let the photometric magnitude, C, be 7.0. 
Then, from the table, x = 0.4, and the magnitudes of A and 
B are 7.4 and 8.2 (to the nearest even tenth). Conversely, 
we may find C from A and B. 

THE OBSERVING PROGRAM 

It has happened in the past that certain well-known double 
stars have been measured and remeasured beyond all reason- 
able need, while other systems of equal importance have been 
almost entirely neglected. The General Catalogues described 
in the preceding chapter make it comparatively easy for 
observers to avoid such mistakes hereafter. In the light of 



54 THE BINARY STARS 

the knowledge these catalogues give of past observations and 
of the motions in the various systems, the observer who 
wishes his work to be of the greatest possible value will 
select stars which are suited to his telescope and which are in 
need of measurement at a given epoch either because of scar- 
city of earlier measures or because the companion is at a critical 
point in its orbit. For example, I am measuring /3 395 = 82 
Ceti regularly at present because there are no measures of the 
companion in the part of the orbit through which it is now 
moving, and I am watching e Equulei closely because at present 
it is apparently single and the position of the companion at 
the time of its reappearance will practically decide the char- 
acter of the orbit. 

It has often been said that a careful set of measures of any 
pair of stars made at any time is valuable. Granting this to 
be so, it is certain that its value is greatly enhanced if it is 
made to contribute to the advancement of a program having 
a definite end in view. If the aim is to increase the number 
of known orbits as rapidly as possible, attention should be 
centered upon the closer pairs, particularly those under 0.5" 
and those which have already been observed over considerable 
arcs of their orbits. I am emphatically of the opinion that 
it is wise for an observer possessing the necessary telescopic 
equipment to devote his energy to the measurement of a 
limited number of such systems, repeating the measures every 
year, or every two or five years, as may be required by the 
rapidity of the orbital motion, for a long series of years. Such 
a series can be investigated for systematic as well as acciden- 
tal errors of measure far more effectively than an equal num- 
ber of measures scattered over a much larger program, and 
will add more to our real knowledge of the binary systems. 
The wider pairs, and particularly those in the older catalogues, 
now need comparatively little attention, so far as orbital mo- 
tion is concerned. Even moderately close pairs, with distance 
from i" to 5", need, in general, to be measured but once in 
every ten or twenty years. Useful programs, however, may 
be made from wider pairs for the detection of proper motions, 
or for the determination of the relative masses in binary sys- 



THE BINARY STARS 55 

terns by means of measures connecting one of the components 
with one or more distant independent stars. Photography is 
well adapted to such programs. 

It is hardly necessary to add that an hour in the dome on 
a good night is more valuable than half a dozen hours at the 
desk in daylight. Everything possible should therefore be 
done to prevent loss of observing time. In this connection I 
have found charts based on the Durchmusterung invaluable 
for quick identification of stars. 

THE RESOLVING POWER OF A TELESCOPE 

It has been shown that the diffraction pattern of the image 
of a point source of light, like a star, formed by a lens "is a 
disk surrounded by bright rings, which are separated by cir- 
cles at which the intensity vanishes."^ 

Schuster gives the formula 

p = m — (5) 

in which p is the radius of a circle of zero intensity (dark ring), 
D the diameter of the lens,/ its focal length, X the wave length 
of the light from the point source, and m a coefficient that 
must be calculated for each ring. For the first dark ring it is 
1.220, and the values for the successive rings increase by very 
nearly one unit. Nearly all of the light (0.839) is in the cen- 
tral disk, and the intensity of the bright difi*raction rings 
falls off very rapidly. Now it is generally agreed that the 
minimum distance at which a double star can be distinctly 
seen as two separate stars is reached when the central disk of 
the image of the companion star falls upon the first dark ring 
of the image of the primary, and the radius of this ring, ex- 
pressed in seconds of arc, is therefore frequently called the 
limit of the telescope's resolving power. If we adopt for X 
the wave length 5,500 Angstrom units, the expression for pin 
angular measure becomes 

P=-^ (6) 

' Schuster, Theory of Optics (1904), p. 130. 



56 THE BINARY STARS 

from which the resolving power of a telescope of aperture D 
(in inches) may be obtained. For the thirty-six-inch Lick 
refractor, the formula gives 0.14", for the twelve-inch, 0.42". 

It will be observed that the resolving power as thus derived 
rests partly upon a theoretical and partly upon an empirical 
basis. When the central disk of each star image of a pair falls 
upon the first dark ring of the other image, the intensity curve 
of the combined image will show two maxima separated by a 
distinct minimum. When the disks fall closer together this 
minimum disappears, the image becomes merely elongated, 
perhaps with slight notches to mark the position of the dis- 
appearing minimum. The pair is now no longer 'resolved' 
according to the definition given but to the experienced 
observer its character may still be unmistakable. For ex- 
ample, in the Lick Observatory double star survey, Hussey 
and I have found with the thirty-six-inch at least 5 double 
stars with measured distances of o.ii" or less, the minimum 
for each observer being 0.09"; and we have found many pairs 
with the twelve-inch telescope whose distances, measured 
afterward with the thirty-six-inch, range from 0.20" to 
0.25". In all these cases the magnitudes were, of course, 
nearly equal. 

Lewis has published * a very interesting table of the most 
difficult double stars measured and discovered by various ob- 
servers using telescopes ranging in aperture from four inches 
to thirty-six inches. He tabulates in separate columns the 
values for the 'bright' and 'faint' pairs of nearly equal magni- 
tude, and for the bright and faint pairs of unequal magnitude, 
each value representing the mean of 'about five' of the closest 
pairs for a given observer and telescope. A final column 
adds the theoretical resolving power derived, not from the 
equation given above, but from Dawes's well-known empirical 
formula — resolving power equal 4.56" divided by the aperture 
in inches (a) — which assumes the two stars to be of about 
the sixth magnitude. Lewis finds that, in general, this for- 
mula gives values which are too small even for the 'bright equal 

♦ The Observatory, vol. xxxvii, p. 378, 1914. 



THE BINARY STARS 57 

pairs', and he suggests the following as representing more 
precisely the results of observation : 

4.8" 

Equal bright pairs , mean magnitudes 5.7 and 6.4 

a 

8 s" 

Equal faint pairs — '■ , mean magnitudes 8.5 and 9.1 

a 

Unequal pairs — '■ — , mean magnitudes 6,2 and 9.5 

a 

Very unequal pairs — '■ — , mean magnitudes 4.7 and 10,4 
a 

Lewis is careful to state that his table does not necessarily 
represent the minimum limits that may be reached with a 
given telescope under the best conditions, and I have just 
shown that they do not represent the limits actually reached 
at the Lick Observatory. Taking from each of the three lists 
of new double stars /8 1,026 to jS 1,274, Hu i to Hu 1,327, and 
A I to A 2,700, 'about five' of the closest bright, and closest 
faint, equal pairs discovered by each of the three observers, 
Burnham, Hussey, and Aitken — 29 pairs in all — I find the 
following formulae for the thirty-six-inch telescope: 

Equal bright pairs — ^ — , mean magnitudes 6.9 and 7.1 
a 

61" 
Equal faint pairs — '■ , mean magnitudes 8.8 and 9.0 

a 

The most interesting point about these formulae is that 
they show much less difference between the values for faint 
and bright pairs than Lewis's do. 

While it is a matter of decided interest to compare the limits 
actually attained with a given telescope with the theoretical 
resolving power, an observer, in making out his working pro- 
gram for double star measurement, will do well to select 
pairs that run considerably .above such limiting distances. 
My deliberate judgment is that, under average good observing 
conditions, the angular separation of the pairs measured should 
be nearly double the theoretical limit. Observers with the 
most powerful telescopes, however, are confronted with the 



58 THE BINARY STARS 

fact that if they do not measure the very closest known pairs 
these must go unmeasured. 

EYE-PIECES 

The power of the eye-piece to be used is a matter of practical 
importance, but one for which it is not easy to lay down spe- 
cific rules. The general principle is — use the highest power the 
seeing will permit. When the seeing is poor, the images 'dan- 
cing' or 'blurred', increase in the magnifying power increases 
these defects in the images and frequently more than offsets 
in this way the gain from increase in the scale. On such 
nights, if they are suitable for any work, choose wider pairs 
and use lower powers. The practical observer soon realizes 
that it is not worth while to measure close pairs except with 
high powers. With the thirty-six-inch telescope my own 
practice is to use an eye-piece magnifying about 520 diameters 
for pairs with angular separation of 2" or more. If the dis- 
tance is only i ", I prefer a power of i ,000, and for pairs under 
0.5", I use powers from 1,000 to 3,000, according to the angular 
distance and the conditions. Closeness and brightness of the 
pair and the quality of the definition are the factors that de- 
termine the choice. Very close pairs are never attempted un- 
less powers of i ,500 or higher can be used to advantage. 

The simplest method of measuring the magnifying power of 
an eye-piece in conjunction with a given objective is to find 
the ratio of the diameter of the objective to that of its image 
formed by the eye-piece — the telescope being focused and 
directed to the bright daylight sky. Two fine lines ruled on a 
piece of oiled paper to open at a small angle form a convenient 
gage for measuring the diameter of the image. A very small 
error in this measure, however, produces a large error in the 
ratio and the measure should be repeated many times and the 
mean result adopted. 

DIAPHRAGMS 

It is sometimes said that the quality of star images is im- 
proved by placing a diaphragm over the objective to cut down 
its aperture. I question this. It is certain that the experience 



THE BINARY STARS 59 

of such observers as Schiaparelli and Burnham is directly 
opposed to it, and experiments made with the twelve-inch and 
thirty-six-inch telescopes offer no support for it. Indeed, it 
is difficult to understand how cutting ofif part of the beam of 
light falling upon an object glass of good figure can improve 
the character of the image, unless it is assumed that the ampli- 
tude of such atmospheric disturbances as affect the definition 
is small enough to enter the problem. The only possible gain 
might be in the reduction of the brightness of the image when 
one star of a pair is exceptionally bright, as in Sirius; but this 
reduction can be effected more conveniently by the use of 
colored shade glasses over the eye-piece. These are occasion- 
ally of advantage. 

A hexagonal diaphragm placed over the objective, however, 
may prove of great value in measuring stars, like Sirius or 
Procyon, which are attended by companions relatively very 
faint; but this is because such a diaphragm entirely changes 
the pattern of the diffraction image of the star, not because 
it cuts down the aperture of the telescope. The pattern is 
now a central disk from which six thin rays run ; between these 
rays the field appears dark even close to the bright star, and 
a faint object there can be seen readily that would be invisible 
otherwise. Professor Barnard ^ has used such a diaphragm to 
advantage with the forty-inch Yerkes refractor. Provision 
should be made for rotating the diaphragm through an angle 
of about 60° and it will be convenient in the case of a large 
instrument to be able to do this by means of gearing attached 
to a rod running down to the eye-end. 

ERRORS OF OBSERVATION 

All measures of angles or of distances are affected by errors, 
both accidental and systematic, and when, as in double star 
work, the measured quantities are very minute, these errors 
must be most carefully considered. The accidental errors 
may be reduced by careful work and by repeating the mea- 
sures a suitable number of times. Little is to be gained, in this 
respect or in any other, by making too large a number of set- 

' A. N., vol. clxxxii, p. 13, 1909. 



6o THE BINARY STARS 

tings upon an object on any one night; because such factors 
as the seeing, the hour angle, the observer's physiological con- 
dition, all remain nearly constant. As a rule, four settings for 
position angle and three or even two measures of double dis- 
tance are enough to make on one night, but the measures 
should be repeated on one or more additional nights. This is 
not only to reduce the accidental error of measure but to 
guard against outright mistakes in reading the circles, record- 
ing, etc. As to the number of nights on which a system should 
be measured at a given epoch, opinions will differ. Some ob- 
servers run to excess in this matter. Generally, it may be said 
that it is time wasted to measure a system on more than four 
nights at any epoch and ordinarily the mean of three nights' 
measures, or even of two, if the pair is easy to measure and the 
measures themselves are accordant, is as satisfactory as the 
mean from a larger number. In critical cases, however, a 
larger number is sometimes desirable. 

The systematic errors of measurement are far more trouble- 
some, for they vary not only with the individual but are dif- 
ferent for the same observer at different times and for different 
objects. Aside from the personality of the observer, they 
depend upon the relative magnitudes of the two components 
of a double star, the angular distance, the angle which the line 
joining the stars makes with the horizontal, and, in unequal 
pairs, upon the position of the faint star with respect to the 
bright one. Various methods have been adopted to deter- 
mine these errors or to eliminate them. 

The most elaborate investigation in this line is probably 
that made by Otto Struve, who measured "artificial double 
stars formed by small ivory cylinders placed in holes in a black 
disk." He deduced formulae by means of which he calculated 
corrections to be applied to all of his measures; but it is very 
doubtful whether these corrections really improve the results. 
I agree with Lewis when he says, "I would prefer his original 
measures — in part because the stars were so particularly arti- 
ficial." The actual conditions when observing the stars at 
night are of necessity widely different from those under which 
the test measures were made. Certainly, in the case of Otto 



THE BINARY STARS 



6i 



Struve, the 'corrected' angles and distances are frequently 
more at variance with the general run of all of the measures 
by good observers than the original values. The student of 
double star motions will generally find it advantageous to use 
the original uncorrected measures of every observer in his pre- 
liminary work and then to derive values for the systematic 
or personal errors of each by comparing his measures with 
the curve representing the means of all available measures. 

The observer, on the other hand, may profitably adopt 
observing methods designed to eliminate, in part at least, 
systematic errors. Innes's plan of measuring each pair on 
each side of the meridian is an excellent one because, in gen- 
eral, the line joining the two stars changes its angle with respect 
to the horizon in passing the meridian. In the extreme case, 
if the smaller star is above the primary when the pair is east 
of the meridian, it will be below, when west of the meridian. 
When Innes's two measures made in this way are not 
sufficiently accordant, he repeats them on two additional 
nights, one night in each position of the instrument. 

In 1908, MM. Salet and Bosler ^ published Jthe results of an 
investigation of the systematic errors in measures of position 
angle in which they made use of a small total reflecting prism 
mounted between the eye-piece and the observer's eye and 
capable of being rotated in such manner as to invert the field 
of view. Theoretically, the half sum of the measures made 
without and with the prism should represent the angle freed 
from errors depending upon the inclination of the images to 
the horizon. In fact, Salet and Bosler found that, whereas 
their measures without the prism and those made with it both 
showed a personal equation varying in amount with the star, 
the means of the two sets were remarkably free from person- 
ality. Here, for example, are their measures of 7 Leonis: 



Observer 


Date 


Without Prism 


With Prism 


Mean 


Salet 

Bosler 

(S-B) 


1907.19 
1907.23 


119.04° 
116.80° 
+2.24° 


113.50° 
116.07° 

-2.57° 


116.27° 
116.44° 
-0.17° 



• Bulletin Astronotnique, Tome 



p. 18, 1908. 



62 THE BINARY STARS 

Hermann Struve and J. Voiite have since published mea- 
sures made in this manner and each concludes that the results 
are far better than his meaures made entirely without the use 
of the prism. In M. Votlte's last paper ^ the statement is 
made that "it is principally in observing in the perpendicular 
(:) position that the observations show a pronounced syste- 
matic error," while "the parallel ( . . ) observations are in gen- 
eral free from systematic errors." 

Dawes ^ long ago pointed out that in "rather close double 
stars," the measures of distance "will almost inevitably be 
considerably too large" unless the observer has taken into 
account the change made in the apparent form of the star 
disk when a thread of the micrometer is placed over it. 
This change is in the nature of a swelling out of the disk 
on each side of the thread, producing an approximately 
elliptical disk. When two images are nearly in contact and 
the threads are placed over them this swelling obliterates 
the interval between the disks and the threads are therefore 
set too far apart. The effect disappears when the disks are 
well separated. 

In my investigations of double star orbits I have frequently 
noticed that distance measures of a given system made with 
small apertures are apt to be greater than those made with 
large telescopes even when made by the same observer, pro- 
vided the system is a close one as viewed in the smaller instru- 
ment. I have found such a systematic difference in the dis- 
tances in stars which I have measured with the twelve-inch 
and with the thirty-six-inch telescope, and Schlesinger ^ has 
also called attention to this difference, giving a table derived 
from my measures as printed in Volume XII of the Publi- 
cations of the Lick Observatory. This table is here reproduced 
with a column of differences added : 

''Circular No. 27, Union Observatory, South Africa, 191 S. 
• Mem. Royal Astronomical Society, vol. xxxv, p. I53i 1867. 
'Science, N. S., vol. xliv, p. S73t 1916. 



THE BINARY STARS 
Measured Separations 



63 



Number of 
Stars 


With the 
12-inch 


With the 
36-inch 


Difiference 


20 


0.52" 


0.42" 


-fo.io'' 


25 


0.62" 


0.54" 


+0.08*' 


20 


0.71'' 


0.64" 


+0.07' 


24 


0.81" 


0.79" 


+0.02" 


24 


1.07^ 


1.03'' 


+0.04*' 


21 


1.38" 


1.39' 


— O.OI" 


26 


2.13'' 


2.10" 


+0.03" 


18 


449" 


4.53" 


— 0.04' 



The systematic difference is clearly shown in all the pairs 
having a separation less than twice the resolving power (042") 
of the twelve-inch telescope ; in the wider pairs it is negligibly 
small. 

Occasionally an observer's work shows systematic differ- 
ences of precisely the opposite sign. Thus Schlesinger (/. c.) 
shows that in Fox's recent volume ^^ of double star observa- 
tions the distances are measured smaller with the twelve-inch 
than with the eighteen and one-half-inch or with the forty- 
inch, "the differences being largest for small separations and 
becoming negligibly small for separations in the neighborhood 
of 5"." The personal equation revealed in such comparisons 
as these must obviously be taken into account in orbit com- 
putations. 

PHOTOGRAPHIC MEASURES 

Photographic processes of measurement are coming more 
and more into favor in almost all lines of astronomical work, 
and with the constant improvements that are being made in 
the sensitiveness and fineness of grain in the plates it is prob- 
able that important work in double star measurement will 
soon be undertaken photographically. Indeed, experiments 
in this line date back to 1857, when G. P. Bond secured with 
an eight-second exposure on a collodion plate the first mea- 
surable images of a double star — f Ursae Majoris, angular 

^^ Annals of the Dearborn Observatory, Northwestern University, vol. i, 191S. 



64 THE BINARY STARS 

separation 14.2". Pickering and Gould in America, MM. 
Henry in France, and the Greenwich observers in England, 
among others, followed up this early attempt and succeeded 
in securing results of value for some stars as close as i ". More 
recently Thiele, Lau, and Hertzsprung at Copenhagen have 
carried out more extensive programs and have investigated 
several possible sources of systematic error in the measure- 
ment of photographic plates. Hertzsprung is at present con- 
tinuing his researches at Potsdam, and finds it possible to 
secure excellent measures of pairs as close as i ". 

There are obvious limitations to the application of photog- 
raphy to double star measurement; very close pairs and pairs 
with moderate distances in which one component is relatively 
faint will not give measurable images on plates at present 
available. On the other hand there is no question but that 
wider pairs can be as accurately, and far more conveniently 
measured photographically than visually, provided systematic 
errors are eliminated. The discovery of faint double stars 
with distances exceeding, say 3^, may also with advantage be 
left to the photographic observer. Comparison under the 
blink microscope of plates taken at suitable intervals with 
instruments giving fields on the scale of the Carte-du-ciel charts 
will quickly reveal any such pairs which show appreciable 
motion and these are the only faint pairs that need be taken 
into serious account in the present stage of double star astron- 
omy. Instruments giving photographs of larger scale will, of 
course, reveal closer pairs. Thus, Fox, on plates exposed with 
the eighteen and one-half-inch Dearborn refractor, has recently 
found two pairs (Fox 11 and Fox 25) with measured distances 
of 1.7" and 1.2", the magnitudes being 9.9 — ii.o and 9.6 — 10.2 
respectively. 



CHAPTER IV 

THE ORBIT OF A VISUAL BINARY STAR 

We have seen that Sir William Herschel, by his analysis of 
the observed motion in Castor and other double stars, demon- 
strated that these systems were "real binary combinations of 
two stars, intimately held together by the bonds of mutual 
attraction." Later observation has shown that the apparent 
motion in such systems is on the arc of an ellipse and that the 
radius vector drawn from the primary star to its companion 
sweeps over areas which are proportional to the times. It has 
therefore been assumed from the beginning that the attractive 
force in the binary star systems is identical with the force of 
gravitation in our solar system, as expressed by Newton's law, 
and the orbit theories which we are to investigate in the present 
chapter are all based upon this assumption. Before taking 
up the discussion of these theories it is pertinent to inquire 
whether the fundamental assumption is justified. 

It is supported by all of the available evidence but rigorous 
mathematical proof of its validity is difficult because the 
motion which we observe in a stellar system is not the true 
motion but its projection upon a plane perpendicular to the 
line of sight. The apparent orbit is therefore, in general, not 
identical with the true orbit and the principal star may lie at 
any point within the ellipse described by the companion and 
not necessarily at either the focus or the center. Hence, in 
Leuschner's words, "mathematical difficulties are encountered 
in establishing a law of force which is independent of the angle 
B, the orientation." In the article quoted, Leuschner, after 
pointing out that "Newton did not prove the universality of 
the law of gravitation, but by a happy stroke of genius gener- 
alized a fact which he had found to be true in the case of the 
mutual attraction of the Moon and the Earth," proceeds to 
show that the law does hold throughout the solar system, the 



66 THE BINARY STARS 

question of orientation not entering. He then says that, in 
binary systems, "when the law is arbitrarily assumed to be 
independent of the orientation, as was found to be the case in 
the solar system, two possibilities arise, namely, either that 
the force is in direct proportion to the distance r between the 
two stars or that the Newtonian law applies. It can be shown, 
however, that when, in the case of an elliptic orbit, the force 
is proportional to r, the primary star must be in the center of 
the ellipse. As this has never been found to be the case, the 
only alternative is the Newtonian law." 

It should be clearly understood that the difficulty in demon- 
strating the universality of the law of gravitation here pointed 
out is purely mathematical. No physical reason has ever been 
advanced for a dependence of an attracting central force upon 
the orientation, and until such dependence has been proved 
we may safely proceed with our investigation of binary star 
orbits under the action of the law of gravitation. 

Until the relative masses of the two components are known 
it is impossible to determine the position of the center of 
gravity of the system and we are therefore unable to compute 
the orbits described by the two stars about that center. What 
our measures give us is the apparent orbit of one star, the 
companion, described about the other, the primary, which is 
assumed to remain stationary at the focus. It is clear that 
this relative orbit differs from the actual orbits of the two com- 
ponents only in its scale. 

The problem of deriving such an orbit from the micrometer 
measures of position angle and distance was first solved by 
Savary,^ in 1827, but Encke ^ quickly followed with a different 
method of solution which was somewhat better adapted to 
the needs of the practical astronomer, and Sir John Herschel ^ 
communicated a third method to the Royal Astronomical 
Society in 1832. Since then the contributions to the subject 
have been many. Some consist of entirely new methods of 
attack, others of modifications of those already proposed. 
Among the more notable investigators are Villarceau, Madler, 

1 Savary, Conn, des Temps, 1830; Encke, Berlin Jahrbuch, 1832; Herschel, Memoirs 
of the Royal Astronomical Society, s, 171, 1833. 



THE BINARY STARS 67 

Klinkerfues, Thiele, Kowalsky, Glasenapp, Seeliger, Zwiers, 
Howard, Schwarzschild, See, and Russell. 

The methods of Savary and Encke utilize four complete 
measures of angle and distance and, theoretically, are excellent 
solutions of the problem; Herschel's method is designed to 
utilize all the available data, so far as he considered it reliable. 
This idea has commended itself to all later investigators. 
Herschel was convinced, however, that the measures of dis- 
tance were far less trustworthy than those of position angle, 
and his method therefore uses the measures of distance simply 
to define the semi-major axis of the orbit; all of the other 
elements depend upon measures of position angle. At the time 
this may have been the wisest course, but the distance mea- 
sures of such early observers as W. Struve, Dawes, and Dem- 
bowski, and those of later observers working with modern 
micrometers, are entitled to nearly or quite as much weight as 
the measures of position angle and should be utilized in the 
entire orbit computation. 

Whatever method is adopted it is clear that the problem 
consists of two distinct parts: first, the determination of the 
apparent ellipse from the data of observation; secondly, the 
derivation of the true orbit by means of the relations between 
an ellipse and its orthographic projection. 

THE appare;^t ellipse 

Every complete observation of a double star supplies us with 
three data: the time of observation, the position angle of the 
companion with respect to the primary, and the angular dis- 
tance between the two stars. It is clear, as Comstock pointed 
out many years ago, that the time of observation is known 
with far greater accuracy than either of the two coordinates 
of position. The relations between the times of observation 
and the motion in the ellipse should therefore be utilized ; that 
is, the condition should be imposed that the law of areal veloc- 
ity must be satisfied as well as the condition that the points of 
observation should fall approximately upon the curve of an 
ellipse. Elementary as this direction is, it is one that has been 
neglected in many a computation. 



68 THE BINARY STARS 

Theoretically, the first step in our computation should be the 
reduction of the measured coordinates to a common epoch by 
the application to the position angles of corrections for pre- 
cession and for the proper motion of the system. The distance 
measures need no corrections. Practically, both corrections 
are negligibly small unless the star is near the Pole, its proper 
motion unusually large, and the time covered by the observa- 
tions long. The precession correction, when required, can be 
found with sufBcient accuracy from the approximate formula 

A0= —0.0056° sin a sec 5 {t—to). (i) 

The formula for the correction due to the proper motion of 
the system is 

A^ = /x"sin 5 (/-/o) (2) 

where m" is the proper motion in right ascension expressed in 
seconds of arc. 

When the measures of any binary star have been tabulated 
(with the above corrections, if required) they will exhibit 
discordances due to the accidental and systematic errors of 
observation and, occasionally, to actual mistakes. If they were 
plotted, the points would not fall upon an ellipse but would 
be joined by a very irregular broken line indicating an ellipse 
only in a general way. It will be advisable to investigate the 
measures for discordances before using them in the construc- 
tion of the apparent ellipse and the simplest method is to plot 
upon coordinate paper first the position angles and then the 
distances, separately, as ordinates, against the times of obser- 
vation as abscissae, using a fairly large scale. Well determined 
points (for example, a point resting upon several accordant 
measures by a skilled observer and supported by the preceding 
and following observations) may be indicated by heavier 
marks. Smooth free-hand curves, interpolating curves, are 
now to be drawn to represent the general run of the measures 
and in drawing these curves more consideration will naturally 
be given to the well observed points than to the others. Obser- 
vations which are seriously in error will be clearly revealed and 
these should be rejected if no means of correcting them is 



THE BINARY STARS 69 

available. The curves will also show whether or not the 
measures as a whole are sufficiently good to make orbit com- 
putation desirable. 

If the amount of available material warrants it, the question 
of the systematic or personal errors of the observers should 
also be considered at this time. No reliable determination of 
such errors is possible unless (a) measures by the same observer 
under essentially the same conditions in at least four or five 
different years are at hand, and (b) unless the total number of 
measures by many different observers is sufficient to establish 
the general character of the curves beyond reasonable question. 
If the second condition is satisfied, the average of the residuals 
from the curve for a given observer may be regarded as his 
personal error and the corresponding correction may be applied 
to all of his measures. Two further points should be noted: 
Firsty the residuals in position angle should be reduced to arc 
by multiplying by the factor p ^ 57.3 before the mean is taken, 
to allow for the effect of variations in the angular separation ; 
second, the corrections should not be considered as constant 
over too long a period of time. Many computers take no 
account of the personal errors in their calculations, and if the 
object is merely to obtain a rough preliminary orbit this prac- 
tice is perhaps legitimate. 

After all corrections have been applied, the measures which 
are retained should be combined into annual means or into 
mean places at longer or shorter time intervals according to 
the requirements of the particular case. Several factors really 
enter into the question of the weights to be assigned to the 
individual observations in forming these means; for instance, 
the size of the telescope used, the observing conditions, the 
number of nights of observation, and the experience of the 
observer; but it will be wise, in general, to disregard all but 
the number of nights of observation, provided the telescope 
used is of adequate resolving power for the system in question 
and that the observer has not specifically noted some of his 
measures as uncertain. A single night's measure deserves 
small weight; mean results based upon from two to six nights' 
accordant measures may be regarded as of equal weight; means 



70 THE BINARY STARS 

depending upon a much larger number of measures may be 
weighted higher. In general, a range in weights from one to 
three will be sufficient. 

Having thus formed a series of normal places, we may find 
the apparent ellipse which best represents them either graphi- 
cally or by calculating the constants of the general equation 
of the ellipse with the origin at any point. This equation is 

ax^-\-2hxy-\-by^-\-2gx-]-2fy+c = o (3) 

which may be written in the form 

Ax^-{-2Hxy+Bf-\-2Gx-\-2Fy+i=o (4) 

in which we must have A>o, B>o, and AB — IP>o. 

If we assume the position of the primary star as origin we 
may calculate the five constants of this equation from as many 
normal places by the relations 

y = P cos u 

but it is advisable to make a least squares solution using all of 
the normal places. 

The great objection to this method is that it entirely disre- 
gards the times of observation. Moreover, the errors of obser- 
vation, small as they are numerically, are large in proportion 
to the quantities to be measured, a fact which makes it difficult 
to obtain a satisfactory ellipse without repeated trials. The 
graphical methods are therefore to be preferred. 

The simplest method, and one that in most cases is satis- 
factory, is to plot the positions of the companion star in polar 
coordinates, the primary star being taken as the origin. With 
the aid of an ellipsograph or by the use of two pins and a 
thread, an ellipse is drawn through the plotted points and is 
adjusted by trial until it satisfies the law of areas. This adjust- 
ment must be made with the greatest precision and the curve of 
the ellipse drawn with great care, for the construction of the 
apparent ellipse is the critical part of the entire orbit deter- 
mination. In my own practice I have found that the test for 
the law of areas can be made most rapidly by drawing radii 
to selected points which cover the entire observed arc and 



THE BINARY STARS 71 

measuring the corresponding elliptic sectors with a planimeter. 
The comparison of the areal velocities derived from the dif- 
ferent sectors at once indicates what corrections the ellipse 
requires. With a suitable ellipsograph a new ellipse is quickly- 
drawn and the areas again measured. The process is repeated 
until a satisfactory ellipse has been obtained. 

Some investigators still prefer the mode of procedure in 
constructing the apparent ellipse first suggested by Sir John 
Herschel. An interpolating curve is drawn, in the manner 
described above, for the position angles only, using the mean 
or normal places. If the curve is carefully drawn, smoothly 
and without abrupt changes of curvature, it should give the 
position angle for any particular epoch more accurately than 
the measure at that epoch, for it rests upon all of the measures. 
From this curve read the times corresponding to, say, every 5° 
of angle, tabulate them, and take the first differences. Divid- 
ing these by the common angle difference will give a series of 

dt 
approximate values of -— . But by the theory of elliptic motion 
du 

p2 -— must be a constant and hence p = c ^ \ Therefore a 

dt '^ \ide 

series of relative values of the distance (expressed in any con- 
venient unit) corresponding to every fifth degree of position 
angle can be derived from the table of angles. Now plot the 
points representing corresponding angles and relative dis- 
tances; if the interpolating curve has been correctly drawn 
and read off they will all lie upon the arc of an ellipse. If they 
do not, draw the best possible ellipse among them and use it 
to correct the interpolating curve, repeating the process until 
the result is satisfactory. Finally, convert the relative into 
true distances by comparing those distance measures which 
are regarded as most reliable with the corresponding values in 
the unit adopted in the plot. 

There are at least two objections to this method: First, it 
does not make adequate use of the observed distances; and 
second, when the angle changes rapidly, as it does in many 
systems at the time of minimum apparent separation, it is 
almost impossible to draw the interpolating curve correctly. 



72 THE BINARY STARS 

In my judgment, It is far better to plot directly the normal 
positions given by the observed angles and distances and then 
by the method of trial and error to find the ellipse which best 
represents them and at the same time satisfies the law of areas. 

THE TRUE ORBIT 

After the apparent ellipse has been constructed graphically, 
or from the constants in the equation of the ellipse, it remains 
to derive the elements which define the form and size of the 
true orbit, the position of the orbit plane, the position of the 
orbit within that plane, and the position of the companion 
star in the orbit at any specified time. Some confusion in the 
nomenclature and even in the systems used in defining these 
elements has arisen from the fact that it is impossible to say, 
from the micrometer measures alone, on which side of the 
plane of projection (which is taken as the plane of reference) 
the companion star lies at a given time. In other words, we 
cannot distinguish between the ascending and the descending 
node, nor between direct and retrograde motion in the ordinary 
sense. Further, in some systems the observed position angles 
increase with the times, in others they decrease. 

The following system is adopted as most convenient when 
the requirements of the observer of radial velocities are con- 
sidered as well as those of the observer with the micrometer. 
In the details in which it differs from other systems in use, it 
was worked out by Dr. Campbell in consultation with Professor 
Hussey and the present writer. 

Let 

P = the period of revolution expressed in mean solar years. 

r=the time of periastron passage. 
e = the eccentricity, 

o = the semi-axis major expressed in seconds of arc. 

fl = the position angle of that nodal point which lies between o° and 
1 80°; that is, the position angle of the line of intersection of 
the orbit plane with the plane perpendicular to the line of 
sight. Call this merely "the nodal point", disregarding the 
distinction between ascending and descending nodes. 

CO = the angle in the plane of the true orbit between the line of nodes 
and the major axis. It is to be measured from the nodal point 



THE BINARY STARS 73 

to the point of periastron passage in the direction of the com- 
panion's motion and may have any value from 0° to 360°. It 
should be stated whether the position angles increase or de- 
crease with the times. 
i = the inclination of the orbit plane; that is, the angle between the 
orbit plane and the plane at right angles to the line of sight. 
Its value lies between 0° and ±90° and should always carry 
the double sign (dz) until the indetermination has been re- 
moved by measures of the radial velocity. When these are 
available, i is to be regarded as positive (+) if the orbital 
motion at the nodal point is carrying the companion star away 
from the observer; negative, if it is carrying the companion 
star towards the observer. 

The symbol n denotes the mean annual motion of the com- 
panion, expressed in degrees and decimals, measured always in 
the direction of motion. 

There is no difference of opinion in regard to the definition 
of the first four elements; the conventions of taking Q. always 
less than 180° and of counting co (for which many computers 
use the symbol X) always in the direction of the companion's 
motion were first suggested, I believe, by See, and have been 
adopted by Burnham, Hussey, Aitken and others. The defini- 
tion of i (for which some computers write 7) is the usual one, 
but computers, as a rule, do not use the double sign. Many 
also prefer to count the mean annual motion in the direction 
of increasing position angles in all systems, and to consider the 
motion negative when the angles decrease with the times. 

When the elements are known, the apparent position angle 
6 and the angular distance p for the time / are derived from 
the following equations : 

M=36o7P 

M=^i{t-T) = E-esmE 

r =a{i—ecosE) (6) 



tsin)4v= ^ l^ tan^E 
\i-e 



tan {d—Q) = rbtan (I'+co) cos i 
p = r cos {v-\-(jt)) sec {d—Q) 



(7) 



74 THE BINARY STARS 

Equations (6) are the usual ones for elliptic motion, the 
symbols M, E, and v representing respectively, the mean, 
eccentric, and true anomaly, and r the radius vector. Equa- 
tions (7) convert the v and r of the companion in the true 
orbit into its position angle and distance in the projected, or 
apparent orbit. Position angles are generally recorded only to 
the nearest tenth of a degree in orbit computation, hence it is 
sufficiently exact to take the value of E corresponding to a 
given value of M from Astrand's Hiilfstafeln, which hold 
for all values of the eccentricity, or the value of v directly 
from the still more convenient Allegheny Tables, provided the 
eccentricity does not exceed 0.77. If the latter tables are used 
it is convenient to derive the value of r from the equation 

r = a {i—e~) /(i+^cosiO (6a) 

instead of from the third of (6). 

kowalsky's method 

From the many methods which have been devised for deriv- 
ing the elements of the true orbit from the apparent ellipse I 
have selected two to present in detail, Kowalsky's and Zwiers's. 
Both are of very general application and are very convenient 
in practice but there are cases in which both fail. Some of 
these will be discussed on a later page. 

Kowalsky's method ^ is essentially analytical and employs 
the constants of the general equation of the ellipse. 

Ax^ + 2 Hxy + By^-{-2Gx-\-2Fy+i =0. (4) 

This is the equation for the rectangular projection of the 
true orbit, the focus of the true orbit falling upon the position 
of the principal star at the origin of coordinates for the ap- 
parent ellipse. Equation (4) may also be regarded as the 
equation of a right cylinder whose axis coincides with the 
2-axis of the system of coordinates, that is, the line of sight. 
Let this equation be referred to a new system of coordinates, 
x\ y', z' , with the same origin, but with the :!c'-axis directed 

2 First published, according to Glasenapp, in the Proceedings of the Kasan Imperial 
University, 1873. This volume has not been accessible to me. 



THE BINARY STARS 75 

to the nodal point, and the ^''-axis at right angles to it in the 
plane of the true orbit. Our transformation equations are 

x = x' cos 12 — y sin fl cos i-\-z' sin Q, sin i 

y = x' sin fi+v' cos Q. cos i—z' cos 12 sin i (8) 

s= H-y sin ^' +3' cos i 

Substituting these values in (4) and placing 2' = o, we shall 
have the equation of the intersection of the cylinder with the 
plane of the true orbit; that is, the real ellipse. Omitting 
accents the equation now becomes 

A {x cos 12 — 3' sin fl cos iY 

+ 2 i? (x cos 12 — >» sin 12 cos i) {x sin 12 + 3' cos 12 cos i) 
+ 5 (x sin 12 + y cos 12 cos iY + 2 G (x cos 12 — y sin 12 cos i) 
+ 2 F (x sin 12 + 3; cos 12 cos i) -{• i = o. (9) 

Now the equation of the true ellipse referred to the focus 
{i. e., the position of the principal star) as origin is 

^ 4- I = o (10) 

the X-axis coinciding with the major axis of the ellipse. Let us 
turn this axis back through the angle co, to make it coincide 
with the line of nodes, by substituting for x and y, respectively, 
the values xcosco+J'sinco, and — :x;sina)+3'cosw, and equation 
(10) becomes 

(xcoso)-\-ys{no}-{-aey . (— xsinco + ^'coso))'' , . 

-: — ■ — - 4- —1=0. (11) 

a^ ^ b^ 

Equations (11) and (9) are necessarily identical since each 
represents the same ellipse referred to the same origin and the 
same axes; therefore the coefficients of the like powers of x and 
y must be proportional. Let/ be the factor of proportionality. 
Then we shall have: 

/( 1 1 =^cos2co + 5sin2 12 + i7sin2l2 (12) 

\ a^ b^ / 

^ /sm^co _^ cos^coX = (^ si^2^4.5cos212-if sin2l2) cos^i (13) 
\ a^ b^ / 



76 THE BINARY STARS 

/( ) sin2a>=(— ^sin2l2+-Ssin2Q4-2Hcos2n)cos'^i (14) 

\ a^ IP- J 

^ecosco ^c;(,osj2+Fsmfi (15) 
a 

/^^^^ = (-Gsinl2 + Fcosl2)cosi (16) 
a 

/ (6^-1) = + I. (17) 
From (17) we find 

I — e^ 

and hence, introducing the semiparameter, p = lPla, the 
relation 

J = = = , or, a = (17a) 

We may now write (16) and (15) in the forms 

— sin CO = — (Fcosfi— Gsintt) cost 

p (18) 

— cos CO = — (Fsin U-\- G cos 12). 
P 

Twice the product of equations (18) is 

— sin 2 o) = {F^ sin 2 12 — G^ sin 2 12 4- 2 FG cos 2 12) cos i (19) 

and equation (14) may be written 

— sin 2 CO = (— ^ sin 2 12 + 5 sin 2 12 + 2 Hcos 2 12) cos i; (20) 

hence 

(F2 - G2 + ^ - 5) sin 2 n^2iFG-H) cos 2 12 = o. (21) 

Subtracting (13) from (12) and substituting for — — 

its equal — - we have 

g2 

— cos 2 CO = ^ cos2 12 + -B sin2 12 + Hsin 2 12 

^' - {A sin2 12 + 5 cos2 12 - iJ sin 2 12) cos^ I (22) 



THE BINARY STARS 77 

The difference of the squares of the two equations (i8) gives 

another value of — cos 2 co. Equating the two values and 

solving for cos^i we obtain 

2 . _ (P- B) sin'^n + {(P - A) cos^n-^ {FG - H) sin 2 12 .^ 

^^^ *~ {F^- B) cos''n+ {G^ - A) siti^a- {FG - H) sm2d 



It is obvious from the forms of the numerator and denomi- 

N 
nator of this equation that if we put cos^t= — and therefore 

,. D-N D+N ^ „ ^ 

tan^^ = = — -- — 2 we shall have 

N N 

tan2i= 2. (24) 

N 

g2 I 

The first member of (13) may be written— -sin^co — ^^"^ 
the equation 

— sin^o; - — = (A sin^ U -\- B cos^ ^ - H sin 2 12) cos^ i. (25) 

/2 I 

Squaring (16) and substituting for — its equal — we find 

— sin2 03= (F^ cos2 12 + G^ ^1^2 ^ - PG sin 2 12) cos^ i; (26) 

and from (25) and (26) 

— = [(P-5)cos212+(G2-^)sin212-(FG-//)sin2l2]cosn\ (27) 

Substituting the value of cos^i from (23) we have 

— =(P-5)sin212+((7-^)cos212+(FG-H)sin2l2 = iV; (28) 



therefore (24) may be written 
2 tan^ 

^ ~7 



+ i^ =F^^Cp.^A + B). (29) 



78 THE BINARY STARS 

Writing for sin^ 12 and cos^ 12 in (28) the corresponding func- 
tions of 2 12 we find 

-^ = P+G2- U+5) -(P-J5)cos2l2 

^' + (G' - ^) cos 2 12+2 (FG-i/) sin 2 12. (30) 

Therefore from (29) we have 



r 



= {F" - C -\- A - B) COS2Q. - 2 {FG - H) sin 2 12. (31) 



Multiply (31) by sin 2 12 and (21) by cos 2 12 and subtract the 
latter result from the former. Then 

^""'Sin 2 12 = - 2 {FG - m. (32) 



Next multiply (31) by cos 2l2 and (21) by sin 2l2 and add the 
products. We have 

tan^^" 



P' 



cos 2 12 = F2 - G2 + ^ - jB. (33) 



Equations (33), (32), (29), (18), and (17a) define the geo- 
metric elements of the orbit in terms of the known constants 
derived from the measures with the micrometer. 

To complete the solution analytically the period P and the 
time of periastron passage T will be found from the mean 
anomalies M computed from the observations by taking the 
ephemeris formulae on page 73 in the reverse order. Every 
M will give an equation of the form 

-^60° 

M = - — (/- D, or, M = fit + e, where e = - fiT 

and the two unknowns P and T will be computed from all the 
equations by the method of least squares. 

GLASENAPP'S MODIFICATION OF KOWALSKY's METHOD 

In theory, Kowalsky's method leaves nothing to be desired; 
given accurate measures it will lead to definitive results. But 
the measures of a double star, as we know, are affected by 
errors that are at present unavoidable, and, until means shall 



THE BINARY STARS 79 

be devised to eliminate these more completely than we are 
now able to do, it will be more practical to adopt Glasenapp's 
suggestion and derive the five constants of the equation of 
the ellipse by his graphical method. Then we may apply 
Kowalsky's formulae, as before, to find the geometric elements 
of the orbit. 

Glasenapp ^ assumes the apparent ellipse to have been 
drawn. Let its equation be, as before, 

Ax"^ + 2 Hxy + 5/ + 2 Gx + 2 F>' + I = o. (4) 

Put3' = o; then the roots of the resulting equation 

Ax'^-{-2Gx-{-i =0 

will be the abscissae of the points of the orbit on the a£: = axis. 
If we represent these roots by Xi and X2, we have, by the theory 
of equations, 

A= , and G = . (34) 

Xi X2 2 Xi X2 

Similarly, if we put x = o, we have the equation 

By^ + 2Fy-{-i =0, 

whose roots will be the ordinates of the points of the orbit on 
the y = axis. From these we obtain 

B=^,andf=-^^i+^. (35) 

yi yi 2 >'i yi 

Therefore, direct measurement of the distances from the prin- 
cipal star to the intersections of the apparent ellipse with the 
axes of X and y, care being taken to regard the algebraic signs, 
will give the four constants A, B, G, and F. H, the remaining 
constant, is derived from 

Ax-'-\-By^'-i-2Gx-\-2 Fy+i . ^- 

n = . k3^) 

2xy 

Measure the coordinates of several points on the apparent 
ellipse, choosing such as will make the product xy as large as 
possible. Each set, substituted in (36) will give a value of H. 

' Monthly Notices Royal Astronomical Society 40, 276, 1889. 



80 THE BINARY STARS 

The accordance of the separate values will depend upon the 
care with which the ellipse has been drawn, and the mean of 
all the values should be adopted. 

Glasenapp's modifications practically convert Kowalsky's 
analytical method into a graphical one for the values of P and 
7", as well as the constants of the general equation which define 
the purely geometrical elements, may be determined by mea- 
sures of the apparent ellipse. It is most convenient to make 
the measures for P and T with the aid of a planimeter as 
follows : 

The position of the periastron point P is at that end of the 
diameter of the apparent ellipse drawn through the origin 5 
which is nearest the origin, for this diameter is clearly the 
projection of the line of apsides of the true orbit. Having 
determined the constant of areal velocity {c) from the portion 
of the ellipse covered by the observations, we measure the 
areas of two sectors, PSt, and PSt\ where t and /' represent 
observed positions on either side of P. Divide these areas by 
c and apply the quotients with the appropriate signs to the 
times corresponding to t and /'. The two resulting values of 
T, the time of periastron passage, should agree closely. More 
points than two may, of course, be used and the mean of all 
the values for T adopted. Similarly, the area of the entire 
ellipse divided by c gives the value of the revolution period. 
Since all the areas are simply relative it is not necessary to 
know the unit of area. 

ZWIERS'S METHOD 

Many methods have been published that enable the com- 
puter to derive the elements of the true orbit from graphical 
constructions. It is impossible to discuss them all in this 
chapter, and it is, fortunately, unnecessary. The crux of our 
problem is the construction of the apparent ellipse; when this 
has been accomplished, almost any of the methods which have 
been proposed will give satisfactory preliminary elements, 
provided the ellipse is a fairly open one. If it is very narrow 
and greatly elongated, none of the ordinary methods is entirely 



THE BINARY STARS 8l 

satisfactory. I have selected Zwiers's * method for presenta- 
tion here because it is as simple as any and is one which I have 
found very convenient. 

Zwiers assumes the apparent ellipse to have been drawn. 
Since it is the projection of the true orbit, the diameter which 
passes through the primary star's position 5 is the projection 
of the major axis of the true orbit and its conjugate is the 
projection of the minor axis. Further, if P is that extremity 
of the diameter through S which is nearest 5 it will be the 
projection of the point of periastron passage in the true orbit. 
Therefore, letting C represent the center of the ellipse, the 
ratio CS jCP will be the eccentricity, e, of that orbit, since 
ratios are not changed by projection. 

Let K = — , be the ratio of the major to the minor axis 

in the true orbit; then, if all of the chords in this orbit parallel 
to the minor axis are increased in the ratio K:i the ellipse will 
be transformed into Kepler's eccentric circle. Consequently, 
if in the apparent ellipse all ordinates parallel to the conjugate 
diameter, described above, are prolonged in the ratio K:i we 
shall have another conic which may be called the auxiliary 
ellipse. It will evidently be the projection of the eccentric circle. 
The major axis of the auxiliary ellipse will be a diameter of 
the eccentric circle and therefore equal to the major axis of 
the true orbit, and its position will define the line of nodes, 
since the nodal line must be parallel to the only diameter not 
shortened by projection. Designate the semi-major and semi- 
minor axes of the auxiliary ellipse by a and /3 respectively; 
then the ratio /3:a is the cosine of the inclination of the orbit 
plane to the plane of projection. Again, the angle co' between 
the major axis of the auxiliary ellipse and the diameter PSCP' 
of the apparent orbit is the projection of the angle oj, the 
angle between node and periastron in the true orbit. Therefore 

tan w' a , , ^ 

tan CO = = — tan co . (37) 

cos i j3 

* A.N. 139, p. 369, 1896. Professor H. N. Russell independently worked out a methott 
based upon the same geometric concept. A. J. 19, p. 9. 1898. 



82 THE BINARY STARS 

Finally P and T are found by areal measures in the apparent 
ellipse in the manner already described. 

The conjugate diameter required in Zwiers's construction 
may be drawn most easily by first drawing any chord of the 
ellipse parallel to PSCP\ the projected major axis. The 
diameter through the middle point of this chord is the con- 
jugate required. If desired, advantage may also be taken of 
the fact that the conjugate diameter is parallel to the tangents 
to the ellipse at the points P and P'. The rectangular axes 
of the auxiliary ellipse may be found by trial or by the fol- 
lowing construction : 

Let 

X2 3/2 



{ay ' {h'Y 

be the equation of the apparent ellipse referred to its conjugate 
diameters. The equation of the auxiliary ellipse referred to 
the same axes will be 

+ 



{a'Y {h'Y 

The axes are therefore also conjugate diameters of the auxiliary 
ellipse. At the extremity P of the diameter a' {PSCP'), erect 
two perpendiculars, PA and PB, to the tangent to the ellipse 
at this point and make each equal in length to Kb'. Through 
the extremities of the two perpendiculars and the center C of 
the apparent ellipse pass a circle. It will cut the tangent in 
two points, A' and B'. The lines A'C and B'C will give the 
directions of the two rectangular axes required, the major 
axis lying in the acute, the minor axis in the obtuse angle 
between the diameters a and Kb'. 

Instead of actually constructing the auxiliary ellipse it will 
generally be easier to derive the elements directly from mea- 
sures of the apparent ellipse with the aid of simple formulae 
based upon the analytical solution of the construction. Thus: 

Let e, a' and b' again represent respectively, the eccentricity, 
and the projected major and minor axes of the orbit, and let 
Xi and X2 be the position angles of a' and b'. To avoid ambig- 



THE BINARY STARS 83 

uity let Xi be the position angle of the principal star as viewed 
from the center of the apparent ellipse and let xz be so taken 
that ixi—X2) is an acute angle. Also, compute as before, 

K = , and b" = Kh'. Then the relation between the rect- 

Vi -e" 

angular axes 2a and 2j3 of the auxiliary ellipse and the con- 
jugate diameters 2a' and 2h" are given by the equations 

a2 + /32 = a'2 -f 6"2 

ajS = a' b" sin (xi - X2) (38) 

the sine being considered positive. 

The coordinates of any point on the auxiliary ellipse with 
respect to the axes 2 a and 2/3 may be written in the form 
a cos </)', j3 sin 0'. Let a cos (co), /3 sin (w) be the coordinates of 
the extremity of the a'— diameter; then we shall have 

a'2 = a2 cos* (co) + ^ sin* (co) (39) 
and 



tan (oj) = ± ^ / (40) 



in which the sign of tan (co) is to be the same as that of ixi — X2). 
But co', the projection of co is related to (co) by the equation 

tan co' = — tan (co) (41) 
a 

that is (co) = CO and ^ = (xi — co'). 

The angle co obviously may have either of two values differ- 
ing by 180°; that value is to be taken which will make 12 less 
than 180°. 

Zwiers counts all angles in these formulae in the direction 
of increasing position angles. 

The practical procedure may therefore be stated as follows : 
Construct the apparent ellipse and the diameter b' conjugate 

to a'; measure e, a\ b\ xi and X2\ compute K= , — =» 

VI -e*, 

b" = Kb', and find a and j8 from 

(a ± ^Y = a'2 + 6"2 ± 2a' b" sin (xi - x^) 

the sine being taken positive. 



84 



THE BINARY STARS 



Then 
a = a 



COS t = 



tan CO 






the sign of tan co being taken the same as that of (xi — :x:2), and 
of the two values of co that one which makes ft less than i8o°. 
Next we have 

tan o)' = — tan co, ft = (x' — co'), 
a 

and finally deduce the values of P and T from area 
measurement, as in the jao* 

Glasenapp-Kowalsky 
method. 

THE ORBIT OF A 88 

The binary system, 
A 88 (R. A. i8h. 33m. 
9s.; Decl. -3° 17'; mag- 
nitudes, 6.9, 7.1), which 
was discovered with the 
thirty-six-inch telescope 
in 1900, will be used 
to illustrate the orbit 
methods which have 
just been described. All 
of the observations of 
this pair have been 
made by the writer, 
and from them it was 
seen that the period 
of revolution must be 
approximately twelve 
years, for in 191 2 the 
companion was again 




Figure 3. The Orbit of A 88 



THE BINARY STARS 



85 



nearly in the position it occupied at the time of dis- 
covery. The following orbit was computed at that time by 
the Glasenapp-Kowalsky method and is purposely not revised, 
although the observed angular motion since then amounts to 
128°, for it will be of interest to see how closely a preliminary 
orbit of a pair so difficult to measure may be expected to rep- 
resent later observations. The maximum separation of the 
two components is only 0.17". 

The observations to date are given in the first three columns 
of the following table. The fourth column shows the number 
of measures (on different nights) on which each position rests, 
and the last two columns give the residuals, observed minus 
computed position angles and distances. One or two of the 
angle residuals are too large to be at all satisfactory; but, in 
estimating them, the extreme closeness of the pair must be 
kept in mind; a residual of 24° in angle corresponds to a dis- 
placement of the companion of but half the thickness of the 
micrometer thread. 



Measures and Residuals for A 88 



Date 


Angle ° 


Disi." 


n 


0- 


-C 


Ad° 


Ap" 


1900.46 


353-2 


0. 14 


3 


- 1.8 


— O.OI 


1901.56 


338-3 


0.14 


3 


- 0.9 


—0.02 


1902.66 


318. 1 


0. 12 


3 


+ 0.6 


— O.OI 


1903-46 


293-6 


O.II 


4 


- 0.8 


±0.00 


1904- 53 


278.4 


0. 14 


4 


+24.1 


+0.03 


1905- 53 


224.8 


0. 12 


4 


+ 0.1 


— O.OI 


1906.48 


199. 1 


0.13 


4 


- 7-1 


-0.03 


1907.30 


193.5 


0. 14 


I 


- 0.5 


-0.03 


1908.39 


178. 1 


0.15 


3 


+ 3-3 


=to.oo 


1909.67 


150.4 


O.IO 


2 


+ 14-7 


+0.03 


1910.56 


47.0 


O.II 


2 


- 0.7 


+0.03 


1911-55 


18.7 


0.15 


I 


+ 6.9 


+0.01 


1912.57 


356.1 


0.15 


3 


+ 0.8 


±0.00 


1914-54 


333.9 


0.15 


5 


+ 11. 2 


+0.01 


1915-52 


306.4 


0.15 


3 


+ 11. 


■4-0.04 


1916.76 


248.4 


0.14 


2 


- 1-9 


+0.02 


1917.65 


228.1 


0.14 


2 


+ 3-2 


+0.02 



86 THE BINARY STARS 

All of the measures to 19 12 inclusive were plotted, using a 
scale of three inches to o.i", and, after repeated trials, 
the ellipse shown in the diagram was drawn. It represents 
the observation points fairly and satisfies the law of areas 
closely. Applying the Glasenapp-Kowalsky method, we first 
measure the intercepts of the ellipse with the axes of coordi- 
nates, and the coordinates of two selected points for the value 
of H, counting the end of the ic-axis at 0°, and of the ^'-axis at 
90°, positive. The measures are (in inches on the original 
drawing) : 

xi=+4-98, >'i= + i.77» '^a=-2.55, ya=-2M 

X2= -473. ^2= -3.12, Xfe= +3.17. yb= -2.49. 

Therefore we have 

xi X2 = - 23.5554, yiy2= - 5-5224, Xaya = -\- 7.2930, 
»:i + 3C2 = + 0.25, yi + y2= — 1.35. Xbyb= -7-^933* 

Xa^ = 6.5025, ya^ = 8.1796 

Xb^ = 10.0489, yb"^ = 6.2001 

from which to compute the five constants of the equation of 
the ellipse. We find 

. I 

A = = - 0.04245 

5:1X2 

B = — — = -0.18108 

yiy2 

F=-5:i±2?= -0.12223 

2 yi yi 
G=_^l±^'= +0.00531. 

20:1X2 

From these values and the coordinates Xa, ya, we obtain 

H = ^ ^ = + 0.00584, 

2xy 

and, similarly, from the coordinates Xb, yb, 

H= -\- 0.00590, 
and adopt the mean, + 0.00587. 



THE BINARY STARS 87 

Combining these constants, we have, 

FG = — 0.00065; P = + 0.01494; CP = -\- 0.00003; 

- 2{FG - H) = 4- 0.013; F^-G^ + A-B=+ 0.15354; 

P + G2-U+5) = +0.23850. 

The solution of equations (32), (33), (29), (18), and (17a) 
then proceeds as follows: 

1. sin 2 12 8. 1 1528 

, tan^i ^ 

1. cos 2 12 9.18622 

P" 

1. tan 2 12 8.92906 

2 12 4.85° 

12 2.4** 

I. cos 2 12 9.99844 



tan^t 
tan^i 



9.18778 
+ 0.15409 



From (29)-^ + -^ +0.23850 



2 


+ 0.08441 


I 


+ 0.04220 


'•; 


8.62536 


\.p^ 


1-37464 


\.p 


0.68732 


1. tan^i 


0.56242 


1. tani 


0.28121 


i 


62.4° 


cost 


9.66586 



88 



THE BINARY STARS 



logF 


9.08717^ 


logG 


7.72482 


sinfi 


8.62557 


sin 12 


8.62557 


costt 


9.99961 


cos 12 


9.99961 


(i)l. Fcosfi 


9. 08678., 


(3) 1. Gsin 12 


6.35039 


(2) 1. Fsin n 


7-7i264„ 


(4) 1. Gcos 12 


7.72443 


(I) 


— 0.12212 


(^ 


+ 0.00022 


(2) 


— 0.005160 


(4) 


+ 0.005302 


(I) - (3) 


-0.12234 


(2) + (4) 


+ 0.000142 


l.-[(i)-(3)] 


9.08757 


l.-[(2) + (4)] 


6.i5229„ 


cos i 


9.66586 


ip 


0.68732 


ip 


0.68732 






log e sin co 


9 44075 


log e cos 0) 


6.83961 


1. tan CO 


2.60II4n 






OJ 


90.1° 






sin CO 


0.00000 






loge 


9-44075 






e 


0.276 






e" 


0.07618 






I -e' 


0.92382 






1.(1 - e') 


9 96559 






\.p 


0.68732 






P 
I- -= logao.72173 

(I - e^) 






a 


5 . 269 inches 
= 0.176" 







From the diagram it is obvious that the companion passed 
its periastron point between the dates of observation 1909.67 
and 1910.56; but the measures made in 1908 and 1912 were 
regarded as more reliable than these and were accordingly used 
to determine the time of periastron passage. The constant of 
areal velocity (in units of the planimeter scale) had been found 
to be 0.205. Drawing radii to the points P and 1908.39 and 
1912.57, the areas of the two resulting sectors were, in terms 
of the same unit, respectively, 0.34440 and 0.50225. Hence 
the time intervals between these two dates and the date of 
periastron passage were, respectively, +1.68 years, and —2.45 



THE BINARY STARS 



89 



years, giving for T, the two values, 1910.07 and 1910.12. The 
mean, 1910.1, was adopted. The planimeter measures gave as 
the area of the entire ellipse, 2.4848, and the period, 12.12 
years. 

To solve the orbit by Zwiers's method, we begin by drawing 
the axis h' conjugate to a' {PSCP'). Draw the chord cc, par- 
allel to P'CSP and then draw the diameter through its middle 
point. This will be the required conjugate. 

We now measure C5 = o.67, CP = 2.45, a' = 2.445, 6' = 5.050; 
and the angles ^^1 = 92.6°, and X2 = 3.6°. 

The ratio CS'.CP gives at once the value of the eccentricity, 

6 = 0.273, and from this we compute the value of K 



Vi -e" 

(in logarithms) 0.01682. Thence we find Z?" = i^6' = 5.2494. 
The computation then proceeds as follows: 



log a' 


0.38828 


(ay 


5 9780 


log 6' 


0.7201 I 


{by 


27-5562 


log 2 


0.30103 






1. sin(^i — X2) 


9-99993 


{{a'Y+{b'y 


33-5342 


.2a'h''sm{xi—X'^ 


I 40.935 


0? 


27.5600 


2a'h"s\n{xi — x<^ 


25-6653 


^ 


5-9756 


{a'Y + {b"Y 


33-5342 


0? - {a'Y 


21.5820 


(a + ^Y 


59-1995 


W - ^ 


0.0024 


(a - /3)2 


7.8689 


log [0? - {a'Y] 


1-33409 


(a + ^) 


7.6942 


\og[{a'Y-m 


7.38021 


(a-^/3) 


2.8052 


log tan^oj 


3-93388 


* 2a 


10.4994 


log tan CO 


1.96694 


2^ 


4.8890 


.*. CO = 


= 89.4° 


a 


5-2497 


log cos i 


9.66805 


^ 


2.4445 


log tan co' 


1.63499 


log^ 


0.38819 


co' 


88.7° 


log a 


0.72014 


.-. 12 = (xi - 


CO') = 3-9 


log cos i 


9.66805 






.'. i = 


62.25° 






a = a = 


= 5.25 inches 







= 0.175" 



90 THE BINARY STARS 

Assembling the elements we have the following: 
Glasenapp's Method Zwiers's Method 



P= 12.12 years 


12.12 years 


T= 1910.10 


1910.10 


e = 0.276 


0.273 


a = 0.176' 


0.175" 


0) = 269.9° 


270.6° 


i = ± 62 . 4 


±62.25 


12 = 2.4 


3.9 



Angles decreasing with the time. 

In the formulae, all angles are counted in the direction of 
increasing position angles, whereas in the notation given on 
page 72 w is counted from node to periastron in the direction 
0} motion of the companion. Therefore, when as in this system 
the observed position angles decrease with advancing time, the 
value for co derived from the formulae must be subtracted from 
360°. In applying the formulae for computing the ephemeris 
of such a system, the anomalies are counted positive after 
periastron passage and negative before, just as in the case of 
direct motion (angles increasing with the time) ; cos i is counted 
as positive, and the angles {6 — 12) are taken in the quadrant 360° — 
(i;+w). I have found this to be the simplest and most satis- 
factory method of procedure in every case where the angles 
decrease with the time. In orbits with direct motion the value 
of CO is used as given directly by the formulae and the angles 
(0 — 12) are taken in the same quadrant as the angles {v-\-(ji). 

CONSTRUCTION OF THE APPARENT ELLIPSE FROM 
THE ELEMENTS 

It is sometimes desirable to be able to construct the apparent 
ellipse from the elements of the true orbit. This construction 
is easily and quickly effected in the following manner: 

Take the point O, at the intersection of two rectangular 
axes, OX and OF, as the common center of the true and pro- 
jected orbits. Draw the line 012 making an angle equal to 12 
with the line OX, counting from 0°. Lay off the angle co from 



THE BINARY STARS 



91 



the line (Xl, starting from the extremity 12 between 0° and 180° 
and proceeding in the direction of the companion's motion (clock- 
wise, that is, if the position angles decrease with the time, 
counter-clockwise, if they increase with the time). This will 
give the direction of the line of apsides, AOP, in the true orbit. 




Figure 4. The True and Apparent Orbits of a 
Double Star (after See) 



Upon this line lay off OS, equal to ae, the product of the eccen- 
tricity and the semi-axis of the orbit, using any convenient 
scale, and OP and OA, each equal to a. The point S lies be- 
tween and P, and P is to be taken in the quadrant given by 
applying cu to fl as described above. Having thus the major 
axis and the eccentricity, the true ellipse is constructed in the 
usual manner. 



92 THE BINARY STARS 

Now divide the diameter UOQ, of this ellipse into any con- 
venient number of parts, making the points of division symmet- 
rical with respect to O, and draw chords h^h', etc. perpendicular 
to the line of nodes. Measure the segments 6/3, ^h' , etc. and 
multiply the results by cos i. The products will evidently be 
the lengths of the corresponding segments /3&i, J862, etc., in the 
projected ellipse, and the curve drawn through the points hi, 
Ci, di, . . . will be the desired apparent orbit. 

To find the position of the principal star in the apparent 
ellipse' draw through S a line perpendicular to the line of nodes, 
and find its intersection S' with an arc drawn with O as a 
center and a radius equal to OScosi. This is the point required. 
Lines through S' parallel to OX and Y will be the rectangular 
axes to which position angles in the apparent orbit are referred, 
and the position angle of the companion at any particular 
epoch may be obtained by laying off the observed position 
angle. The line OS' extended to meet the ellipse defines P', 
the projection of the point of periastron passage. 

DIFFERENTIAL CORRECTIONS 

If sufficient care is exercised in the construction of the 
apparent ellipse, methods like those described will, as a rule, 
give a preliminary orbit which will satisfy the observed posi- 
tions within reasonable limits and which will approximate the 
real orbit closely enough to serve as the basis for a least 
squares solution. It may be remarked that a satisfactory 
representation of the observed positions does not necessarily 
imply a correct orbit when the arc covered by the observations 
is comparatively small. The percentage of error inherent in 
double star measures in so great that, if the observed arc is 
less than 180°, it will generally be possible to draw several 
very different ellipses each of which will satisfy the data of 
observation about equally well. In general, it is not worth 
while to compute the orbit of a double star until the observed arc 
not only exceeds 180°, but also defines both ends of the apparent 
ellipse. 

Many computers are content with a preliminary orbit; but 
it is advisable to correct these elements by the method of least 



THE BINARY STARS 93 

squares whenever the data are sufficient for an investigation 
of the systematic errors of observation. 

The position angle is a function of the six elements 
fl, i, 0), e (= sin </>), T and jjl = 

and the required differential coefficients for the equations of 
condition can be computed with all necessary accuracy from 
the approximate formula 

aAU + bAi + cAo3 -f dA<t) + eAMo +fAfi + (Co - Oo) = o (43) 

where A12, etc., are the desired corrections to the elements, 
Mo = iJi{t—T), (Co — Oo) is the residual, computed minus ob- 
served position angle, and a, b, c, d, e, f, are the partial differ- 
ential coefficients. 

These are derived from the equations 

M = fi{t- T) = E- esinE 

I I -[- e 
tan i^ y = ^ / ■ — tan >^ £ 

\ I — e 

tan (0 — 12) = cos i tan {v + ^) 
and their values are 

0=1 

& = — sin i tan {v + w) cos^ {d — Q) 

c = cos^ {B — 12) sec^ {v + co) cos i 

2 - e cosE - e^ . 
d = sinii-C 

{i — e cos EY 

COS(/) 



(i — ecosEY 
f = {t-T)'e. 



To facilitate the solution, the coefficients in the equations of 
condition should be reduced to the same numerical order by 
the introduction of suitable multipliers. 

The corresponding differential equation for the distance cor- 
rection may be derived by differentiating the formula 

p = a (i — e COS E) cos (v + co) sec {6 — 12), 



94 THE BINARY STARS 

but it is customary to compute the correction for a, the 
semi-axis major, directly from the residuals in distance after 
the remaining elements have been corrected by the aid of 
equation (43).^ 

Equation (43) is strictly applicable only when the residuals 
in angle are independent of the angular distance between the 
companion and primary star. When the eccentricity and the 
inclination of the orbit are both small this condition is approx- 
imately realized, but when either of these elements is large, it 
is clear that the space displacement produced by a given error 
in angle will vary greatly in different parts of the orbit, and 
the equation must be modified so that the solution will make 
the sum of the squares of the space displacements a minimum 
rather than that of the angle residuals. This can, in general, 
be effected with sufficient accuracy by multiplying the values 

'Comstock has just published (The Orbit of S 2026. By George C. Conjstock. As- 
tronomical Journal, vol. 31, p. 33, 1918) expressions which are more convenient in numerical 
application than those given in the text and which have the additional advantage of per- 
mitting the equations derived for Ap to be combined into a single solution with those for 
Ad after Ac has been eliminated from the one group and Afl from the other. 

Write the two groups in the forms: 

(for 6) A Ai2 + BAoi + CAi + DA<f> + F/iAT + GA/t -ir ijC - 0) =0. 
(for p) hAa + 6Aw + cAi + dA<i> +fnAT -f gA/i + {C - O) =0. 

As in equation (43) the eccentric angle <^, defined by sin = c, is introduced as an 
element instead of the eccentricity e. To make the two groups of equations homogeneous, 
(C — O) in the equations for 6 must be expressed in circular measure, that is, instead of 
Ad we must write pAd -^ 57- "3. The corresponding corrections are taken into account 
in the differential coefficients which follow. For convenience form the auxiliary quantities 

nt{ = plS7°3) = [8 . 2419] p, K = (2 + sin <^ cos v) sin E 
a >= — m tan t sin {,6 — S2) cos (0 — 12). 
Then we have 

A = +w A = + -^ 

B = -\-m (— T-J2cos» b = +crsin* 

C ^ ^-tT c = +o- tan (©- 12) 

D = + B K d = + b —— K — m — — cos cos » 

F = - B (-^Wos0 f = - b (-y-J2cos0-m (-7-)^ sin <t> sin E 

G = -Fit-T) g = -fit -T) 

The solution of these equations gives Ac in seconds of arc and the other unknowns in 
degrees. If 6 decreases with the time, count the anomalies positive before periastron, 
negative cifter periastron passage. 



THE BINARY STARS 95 

(Co — Oo) by factors proportional to the corresponding observed 
distances. 

SPECIAL CASES 

The methods of Kowalsky and of Zwiers and all other 
methods based upon the construction of the apparent ellipse 
fail when the inclination of the orbit plane is 90° ; for then the 
apparent ellipse is reduced to a straight line and the observed 
motion is entirely in distance, the position angle remaining con- 
stant except for the change of 180° after apparent occultation. 
Such a limiting case is actually presented by the system 42 
Comae Berenices, and many other systems are known in which 
the inclination is so high that the apparent orbit is an extremely 
narrow ellipse, differing but little from the straight line limit. 
Special methods, based chiefly upon the curve of the observed 
distances, must be devised in such cases. Advantage may 
also be taken of peculiarities in the apparent motion in some 
systems to obtain approximate values of one or more of the 
orbit elements. 

In addition to the inclination ( = 90°), the element fi is 
known from the conditions in the case of a system like 42 
Comae, for this must be the mean of the observed position 
angles. The remaining elements must be determined from the 
curve representing the observed distances. Let us assume that 
the observed distances have been plotted against the times 
and that the most probable curve has been drawn through the 
plotted points. The revolution period may then be read 
directly from the curve, the accuracy of the determination 
depending upon the number of whole revolutions included in 
the observations as well as upon the precision of the measures. 
In general, the elements e, the eccentricity, and T the time of 
periastron passage, are as easily determined. 

Let PK2 Pi, in Fig. 5, represent the true orbit, and BS'C'B', 
its projection. The point C is known for BC must equal half 
the amplitude of the curve of distances. The point 5' is known, 
since it is the origin from which the distances are measured, 
i. e., the position of the primary star. It is also evident that 
the points on the curve of distances which correspond to the 



96 



THE BINARY STARS 



points P and Pi in the true orbit must be separated by precisely 
half of the revolution period and that their ordinates, measured 
from the line CEE' , must be equal in length and of opposite 
sign. The point corresponding to periastron must lie on the 
same side of this line as S', and on the steeper branch of the 
curve. In practice these two points are most readily found by 
cutting a rectangular slip of paper to a width equal to half that 




Figure 5. Apparent and True Orbits, and Interpolating Curve of Observed 
Distances for a Binary System in which the IncUnation is 90° 

of the period on the adopted scale and sliding it along the 
curve until the edges, kept perpendicular to the line CEE' y cut 
equal ordinates on the curve. 

When P has been found on the curve, draw the line PP' 
parallel to CEE'. The value of e follows at once from the 
ratio C'S': C'P'. 

There remain the two elements a and co; and these cannot 
be derived quite so simply. The following process for their 
determination is due to Professor Moulton.^ 

Let d be the angle between the line of apsides and the line 
to the Earth. It is equal to 90° — oj. Then in the figure, we 
have 

S'C = SE = ae sin 0. (44) 

This gives one relation between the three elements a, e and 6, 
for the length SE is known. 

To find another let us take the equation of the ellipse with 
the origin at its center, assuming the X-axis to be, as usual, 
coincident with the major axis. We have 

' Kindly sent to me by letter. 



THE BINARY STARS 97 

Remove the origin to the focus, S, and the equation becomes 

c2 "^ a2 (I - e') 

which we shall write in the form 

{xi - aey{i - e^) + yi^ - ^2(1 - e^) = o. (45) 

Now rotate the axes backward through the angle 6, thus 
making the X-axis point toward the Earth, the transformation 
equations being 

Xi = X cos 6 -{- y slnO 
yi = X sin 6 -\- y cos 9. 

We obtain 

/ (^13') = (i ~ ^^ cos'^ 6) x^ -\- {i — e^ sin^ 6)y^ — 2e^ sin $ cos 6 xy 
— 2ae (i — 6") cos Ox — 2ae (i — e^)s\n6y — a^ (i — e^Y = o. (46) 

Let y = c, be the equation of a line parallel to the X-axis and 
cutting the ellipse in two points. The oc-coordinates of the 
points of intersection are given by 

/ C'^i c) = (i — e^ cos^ 6) x^ — 26 cos 6 [ec sin d -{■ a {i — e^)] x 

+ (i - e2 sin2 e)c^-2aec{i - e^) sin 6 - a^ {i - e'Y = o. (47) 

To obtain the tangent Ki, Li, we must impose the condition 
that the two roots of equation (47) are equal ; that is, that 

df 

-^ = 2 (i - e2 cos2 e)x-2e cos [ec sin + a (i - e^)] = o. (48) 
dx 

Substitute the value of x from (48) in (47), and solve for e 
which, by the conditions, is equal to EL ( = C'B') and is there- 
fore a known quantity. After simplification we obtain 

0^ — 2 aec sin 6 — a^ {i — e^) = o (49) 

which is the desired second relation between the three elements. 
Combining (44) with (49), we have 

y-2aecs\n~e ^ j jELiY- 2SE'ELi _ kc'B'Y- 2S'C' -CB' 
^~\ I -e^ ~ \ 1-^2 ~ \ 1-^2 (50) 

and the value of co ( = 90° — 0) follows at once from (44). 



98 THE BINARY STARS 

If we please, we may write a third relation, independent of 
the other two since it is dynamical and rests upon the law of 
areas, in the form, 

S'P' = a(i - e) sin d. (50a) 

Now let us write (44), (49), and (50a) in the forms 

ae sin d = A 

a{i — e) sin e = B (50b) 

^2(1 _ g2) _|_ 2ae Csmd = C? 

and we find at once 

A 



e = 



A-\-B 



VC2_ 2AC 
a = - — 



(50c) 



. ^ Vb^-\-2AB 
sinO = 



VC^-2AC 
where A = SE = S'C, B = S'P' and C = ELx = C'B'. 

This solution fails only when the points C and S' are coin- 
cident, that is, when the two elongation distances are equal, 
and this will only occur (i) when the true orbit is circular, or 
(2), when the major axis of the ellipse lies in the line of sight. 
The two cases may be distinguished by the fact that in the 
former the time mtervals from apparent coincidence of the two 
stars to the elongations at either side will be equal, in the 
latter,, unequal. When they are equal, the elongation distance 
is the radius, or semi-axis, a, of the true orbit, and any con- 
venient epoch, for example that corresponding to apparent 
coincidence, may be adopted as origin in reckoning the times. 
The elements T and co have, of course, no significance in this 
case. When the elongation times are unequal, the elongation 
distance gives the semi-axis minor, h, of the true orbit, the 
epoch of coincidence which falls in the shorter interval between 
successive elongations is the epoch of periastron, and w equals 
fl =t 90°. The element e cannot in this case be found by the 
direct method given above. Probably it may be derived from 



THE BINARY STARS 99 

the dynamical relation T1IT2 = f (ai, ei, 6), where Ti and T2 
are the epochs of the two elongations. The case will be a very 
rare one, and I have not attempted its solution. 

When a preliminary set of elements has been derived by 
the methods described, improved values may be computed by 
the method of least squares, the equations of condition being 
obtained by differentiating the formula for the apparent dis- 
tance, which in orbits of this character takes the form 

p = r cos(i^ + co) = a{i — e cos E) (cos v + co). 
We may write the equations of condition in the form 

Ap = AAa + BAco + CA</) + DAMo + D{t - T)Aix (sod) 
where 

a 
B = — r sin(t; + w) 

C = ( — J ( — j (sin<^ — cos E) — sin v sin(i> -}- cu) 

( I -\ — )acos0 

\ a cos^ 9/ 

n W ^\^ /^ P \ .L • TT sini'sin(t; + co)] 

D = a\ [ — I I — 1 sin (b sin E . 

l\ r J \ a J ^ sin£ J 

THE ORBIT OF e EQUULEI 

An excellent example of an orbit whose computation was 
made possible by taking advantage of the special features of 
the observed motion is that of e Equulei, recently published by 
Russell.'^ The apparent orbit of this system is an extremely 
narrow and elongated ellipse. Fortunately the double star was 
discovered by Struve, in 1835, when the angular separation 
was only 0.35". In later years the companion moved out to a 
maximum elongation of 1.05" and then in again until now 
(191 7) the pair cannot be resolved by any existing telescope. 
Plotting the distances (using mean places) against the times, 
Russell noted that the curve was practically symmetrical with 

^ Astronomical Journal 30, 123, 191 7. 



100 THE BINARY STARS 

respect to the maximum separation point. It follows that the 
line of apsides in the true orbit must be approximately coin- 
cident with the line of nodes, or in other words that co = o. 
Further, the mean of the position angles for a few years on 
either side of the time of elongation gives a preliminary value 
for the angle fi, and the elongation time itself is the epoch of 
apastron passage, which may be taken in place of the epoch 
of periastron as one of the orbit elements. It is also apparent 
that the inclination of the orbit is very high and a preliminary 
value for this element may be assumed. This leaves the three 
elements, a, e and P, which Russell finds as follows: 

Let yi = the maximum elongation distance 

T' = the corresponding epoch {i. e., apastron) 

y„ = the distance at any other time t„ 

E = the corresponding eccentric anomaly in the true orbit. 

Then we have 

a(i -\- e) = yu a (cosE — e) = —y„ 

M = E- ecos £, and (t„ - T) = i8o° - M, 

which determine a and P in terms of e. 

Assume values of e and compute a and P, repeating the 
process until those values result which represent the curve of 
the observed distances. 

The preliminary elements obtained by these processes 
Russell corrects differentially, a, e, T and jjl from the observed 
distances, i and 12 from the observed angles, co ( = o) being 
assumed as definitely known. 

SYSTEMS IN WHICH ONE COMPONENT IS INVISIBLE 

Luminosity, as Bessel said long ago, is not a necessary attri- 
bute of stellar mass, and it may happen that one component 
of a double star system is either entirely dark or so feebly 
luminous as not to be visible in existing telescopes. If the 
orbit is one of short period and the inclination of its plane 
sufficiently high, the system may be detected by the spectro- 
scope, by the methods to be discussed in the following chapter. 
In other instances the companion's presence may be revealed 



THE BINARY STARS lOI 

by a periodic variation in the bright star's proper motion, the 
path described by it upon the celestial sphere becoming a 
cycloid instead of the arc of a great circle. A system of the 
latter type is most readily detected when the proper motion 
is large, and it is, of course essential that the motion be accu- 
rately determined. 

Variable proper motion was actually recognized in the stars 
Sirius and Procyon, about three-quarters of a century ago, and 
was explained by Bessel as the effect of the attraction of such 
invisible companions. Orbits, referring the motion of the 
bright star to the center of gravity of a binary system, were 
thereupon computed for these stars by C. A. F. Peters and 
A. Auwers. Bessel's hypothesis was proven to be correct by 
the subsequent discovery of a faint companion to Sirius by 
Alvan G. Clark (in 1861), and of a still fainter companion to 
Procyon by Schaeberle (in 1896). The relative orbit of the 
companion to Sirius has been computed from the micrometer 
measures, and the elements are consistent with those deter- 
mined from the proper motion of the bright star. There is no 
question but that this will also prove to be the case in the 
system of Procyon when the micrometer measures permit an 
independent determination of its orbit. 

Dark companions to /3 Orionis, a Hydrae, and a Virginis 
have also been suspected from supposed irregularities in the 
proper motions, but closer examination of the data has not 
verified the suspicion. Since cases of this kind will probably 
always be very exceptional, the formulae for their investiga- 
tion will not be considered here. Those who are interested in 
their development are referred to the original memoirs.^ 

The presence of invisible companions in several well-known 
double star systems has also been suspected on account of 
observed periodic variations in the motion of one of the visible 
components. In one of these, e Hydrae, the primary star was 
later found to be a very close pair whose components complete 

» Bessel. A. N. 22, 14s, 169, 185. 1845. 
Peters, A. N. 32, i, 17, 33, 49, 1851. 

Auwers, A. N. 63, 273, 1865 and Unlersuchungen iiber verdnderliche Eigenheivegung, 
I Theil. Koningsberg, 1862; 2 Theil, Leipzig, 1868. See also A. N. 129, 185, 1892. 



102 THE BINARY STARS 

a revolution in about fifteen years, and Seeliger^ has shown 
that the orbital motion in this close pair fully accounts for the 
irregularities observed in the motion of the more distant com- 
panion. Another of these systems, f Cancri, consists of three 
bright stars, two of which revolve about a common center in a 
period of approximately sixty years, while the third star re- 
volves with this binary system in a much larger orbit. Seeliger 
has shown that the irregularities observed in the apparent 
motion of this third star may be explained on the hypothesis 
that it is accompanied by an invisible star, the two revolving 
about a common center in circular orbits with a period of 
eighteen years. The system would, then, be a quadruple one. 
There are irregularities in the observed motion of 70 Ophiuchi 
which are almost certainly due to the perturbations produced 
by a third body, but a really satisfactory solution of the orbit 
has not yet been published. Finally, Comstock^" has just pub- 
lished a model investigation of the orbital motion in the sys- 
tem ^ Herculis from which he concludes that small irregularities 
in the areal velocity of the bright pair may be represented as 
the effect of an invisible companion to one component, having 
a periodic time of 18 years and an amplitude less than o.i". 
Comstock, however, points out that when the systematic 
errors of the observers are determined and allowed for, the 
orbit, without the assumption of a third body, "satisfies the 
observations within the limits of error commonly deemed satis- 
factory." The paper is an excellent example of the method in 
which systematic errors should be investigated in the compu- 
tation of a definitive double star orbit. 

It is probable that the invisible companion in such a system 
as that of f Herculis revolves, like the bright components, in 
an elliptic, rather than a circular orbit; and it is not at all 
improbable that the plane of this orbit is inclined at a greater 
or less angle to the plane of the orbit of the visible system. 
To determine the eccentricity and the inclination, however, 
would greatly complicate the problem and the precision of the 
observational^data is not sufficient to warrant such refinements. 

• Asironomische Nachrichten 173, 325, 1906. 
^^ Astronomical Journal 30, 139, 1917. 



THE BINARY STARS I03 

In practice, it has been found satisfactory to assume that the 
invisible body moves in a circle in the plane of the orbit of the 
visible stars of the system. This assumption leaves but two 
elements to be determined, the period and the radius or semi- 
amplitude, and the formulae for these are quite simple. Com- 
stock's formulae for the companion in the system of f Herculis, 
for example, are as follows : 

Let B, p, represent the polar coordinates of the visible com- 
panion referred to the primary star; ^, r the corresponding 
coordinates of the center of gravity of the assumed system 
{i. e. the system comprised of the secondary bright star and 
its dark companion) referred to the same origin ; and v, a, the 
coordinates of the visible companion referred to the center of 
mass of itself and its dark companion. Then we shall have 
from the geometrical relations involved, 

p2 = ;'2 _|_ ^2 _j_ 2ar cos(i' — yp) 

6 =yP-\- —sm{v-rP). (51) 
P 

If we assume that — and — are quantities whose squares 

are negligibly small, we have by differentiation 
dS Jyp 



P' . 

dt dt 



-harcos{v-\p) "^ + "IT | ^ ^^^ (^ ~ '^) ^ • (52) 



Since the assumed system is circular, a and — are constant 

dxP ^^ 

quantities, r^ — is also a constant, and a is so small that, in 
dt 

the second member of the equation, we may write 6 for \f/ and 

p in place of r without sensible error. If, further, for brevity, 

^ dxl/ K . , dv , . , , ^ 

we put — = — and k = — , the equation takes the form 
dt p2 dt 

p^— = K + a(kp+ — ^ cos {v-d)-a sin (v-S)^. (53) 
at \ p / dt 

RECTILINEAR MOTION 

The relative motion in some double stars is apparently recti- 
linear and it is desirable to have criteria which shall enable us 



104 



THE BINARY STARS 



to decide whether this is due to the fact that the orbit is a very 
elongated ellipse, or to the fact that the two stars are un- 
related and are changing 
their relative positions by 
reason of the difference in 
their proper motions. One 
excellent test, which has 
been applied by Lewis to 
many of the Struve stars, 
is that if the stars are un- 
related the apparent motion 
of the companion referred 
to the primary will be uni- 
form whatever the angular 
separation of the stars; but 
if they form a physical sys- 
tem, it will increase as the 
angular distance dimin- 
ishes. 

A more rigorous test is 
the one applied, for example, 
by Schlesinger and Alter ^^ 
to the motion of 6i Cygni. 

If the motion is uniform and in a straight line, the position 
angles and distances of the companion referred to the primary 
may be represented by the equations 




Figure 6. Rectilinear Motion 



0?+ {t- TYm" 



tan(0 



0) = — (/ - D 
a 



(54) 



in which a is the perpendicular distance from the primary, 
considered as fixed, to the path of the companion; (jy is the 
position angle of this perpendicular; T, the time when the 
companion was at the foot of the perpendicular, and w, the 
annual relative rectilinear motion of the companion. Approx- 
imate values for these four quantities may be obtained from 
a plot of the observations and residuals may then be formed 

" Publications Allegheny Observatory 2, 13, 1910. 



THEBINARY STARS 105 

by comparing the positions computed from the formulae with 
the observations. If these residuals exhibit no systematic char- 
acter, rectilinear motion may be assumed; if they show a 
systematic course a closer examination is in order to decide 
whether this is due to chance or to orbital motion. In the 
latter case, the indicated curve must be concave to the primary 
and the systematic run of the residuals should be quite uni- 
form. In any event, a least squares solution may be made to 
obtain more precise values for the quantities a, </>, T and m. 
For this purpose, differentiate equations (54) and introduce the 

values sin {e — 4>)= —^ , cos (d — (j))= — (see Fig. 6) ; we 

P P 

thus obtain the equations of condition in the form given by 
Schlesinger and Alter: 

- cos(^ - 0o)Aa - sin(^ - </)o)(/ - To) Am 

+ sin((? - 4>o)moAT + Ap = f p (55) 

-\- sin(0 — 0o)Aa — cos(0 — 4>o) {t — To)Am 

-f cos(0 - ^o)woAr — pAxf/ + pAd = v0 

n which the subscript o indicates the preliminary values of the 
elements, Ap and Ad the deviations from the approximate 
straight line and Vp and ve the residuals from the definitive 
values of the elements. 

REFERENCES 

In addition to the papers cited in the footnotes to the chap- 
ter, the student of double star orbit methods will find the 
following of interest: 

Klinkerfues. tJber die Berechnung der Bahnen der Doppelsterne. 
Astronomische Nachrichten, vol. 42, p. 81, 1855. 

. Allgemeine Methode zur Berechnung von Doppelsternbahnen. 

Astronomische Nachrichten, vol. 47, p. 353, 1858. 

Thiele. tJber einen geometrischen Satz zur Berechnung von Dop- 
pelsternbahnen — u. s. w. Astronomische Nachrichten, vol. 52, p. 
39, i860. 

. Unders^gelse af Oml0bsbevagelsen i Dobbelstjernesystemet 7 

Virginis, Kj0benhavn, 1866. 



I06 THE BINARY STARS 

Thiele. Neue Methode zur Berechnung von Doppelsternbahnen. 
Astronomische Nachrichten, vol. 104, p. 245, 1883. 

Seeliger. Untersuchungen (iber die Bewegungsverhaltnisse in dem 
dreifachen Stern-system f Cancri. Wien, 1881. 

. Fortgesetzte Untersuchungen tiber das mehrfache Stern- 
system f Cancri. Miinchen, 1888. 

Schorr. Untersuchungen iiber die Bewegungsverhaltnisse in dem 
dreifachen Stern-system ^ Scorpii. Miinchen, 1889. 

ScHWARZSCHiLD. Methode zur Bahnbestimmung der Doppelsterne. 
Astronomische Nachrichten, vol. 124, p. 215, 1890. 

Rambaut. On a Geometrical Method of finding the most probable 
Apparent Orbit of a Double Star. Proceedings Royal Dublin Society, 
vol. 7, p. 95, 1891. 

Howard. A Graphical Method for determining the Apparent Orbits 
of Binary Stars. Astronomy and Astrophysics, vol. 13, p. 425, 1894. 

Hall. The Orbits of Double Stars. Astrophysical Journal, vol. 14, 

p. 91, 1895. 
See. Evolution of the Stellar Systems, vol. i, 1896. 

Leuschner. On the Universality of the Law of Gravitation. Uni- 
versity of California Chronicle, vol. XVHI, no. 2, 1916. 

Andre. Trait6 d'Astronomie Stellaire, vol. 2. 

Also the chapters on double star orbits in such works as Klinkerf ues- 
Buchholz, Theoretische Astronomic; Bauschinger, Die Bahnbestim- 
mung der Himmelskorper; Crossley, Gledhill, and Wilson, A Hand- 
book of Double Stars. 



CHAPTER V 

THE RADIAL VELOCITY OF A STAR 
By J. H. Moore 

The observations treated in the preceding chapters concern 
only that part of the star's actual motion in space, which 
appears as change of position in a plane perpendicular to the 
line joining the observer and star. Of the component directed 
along the 'line of sight', called the star's 'radial motion', the 
telescope alone gives no indication. In fact, the possibility 
of detecting radial motion was recognized less than seventy- 
five years ago, and the methods of its measurement belong 
distinctly to another and newer branch of astronorny, known 
as astrophysics. Moreover, observations of the rate of change 
of position of a star on the celestial sphere can be translated 
into linear units, such as kilometers per second, only if the 
star's parallax is known, while measures of radial velocity 
by the method to be described, are expressed directly in kilo- 
meters per second and are independent of the star's distance. 

The determination of the radial velocity of a light source, 
such as a star, is made possible by two well-known properties 
of light; namely, that it is propagated as a wave motion, and 
with a definite and finite velocity. We are not concerned with 
the properties of the hypothetical medium, called the ether, 
in which these waves move, nor with the nature of the dis- 
turbance in the ether, whether it be mechanical or electromag- 
netic. For our purpose it is sufficient to know that in this 
medium, or in interstellar space, the velocity of light is about 
299,860 kilometers per second, and that the well-known laws 
of wave motion hold for light waves. 

In 1842, Christian Doppler called attention to an effect 
upon the apparent length of a wave which should result from 
a relative motion of the source of the waves and the observer. 
This result was independently reached and further developed, 



I08 THE BINARY STARS 

especially with reference to light waves, some six years later 
by the great French physicist, Fizeau. According to the 
Doppler-Fizeau principle, when the relative motion of the light 
source and the observer is such, that the distance between the 
two is increasing or decreasing, the length of the waves received 
by the observer will be respectively longer or shorter than the 
normal length of these waves. 

It is readily shown that the change in wave-length is directly 
proportional to the normal length of the wave and to the ratio 
of the relative velocity of source and observer to the velocity 
of propagation of the waves. Moreover, the change is the same 
whether the source or observer, or both are moving, providing 
their velocities are small in comparison with that of the waves. 
In the case of light waves, and for the celestial objects with 
which we have to deal, this condition is always fulfilled. 

Let us denote by v the relative radial velocity in kilometers 
per second of a star and observer, where v is considered posi- 
tive when the distance between the two is increasing and 
negative when this distance is decreasing. Call X' the wave 
length of a monochromatic ray reaching the observer, whose 
normal wave-length, as emitted by the star is X. 

Then from the Doppler-Fizeau principle, X' — X :X wv : 
299,860;^ or X' — X = Xz; /299860 (if z; is +, X' is greater than X), 
or, writingAX for the change in wave length (X' — X), we have 
for the relative radial velocity of star and observer 

299860 AX , . 

" X ^'^ 

The determination of the radial velocity of a star rests then 
upon a knowledge of the velocity of light and of the wave- 
lengths of certain definite rays emitted by a source at rest, and 
the measurement of the apparent wave-lengths of those same 
rays received from a star. In short, the problem reduces to one 
of measuringAX with the greatest possible precision. For this 
purpose the micrometer with which we have become familiar, 

1 The velocity of light, in kilometers per second. 



THE BINARY STARS IO9 

is replaced by the spectroscope. This wonderful instrument 
originating in the physical laboratory has developed a whole 
new science, spectroscopy, with an extensive and technical 
literature of its own. In this chapter we shall only call atten- 
tion to some of the elementary principles of spectroscopic 
analysis and give a very brief survey of the spectrographic 
method as applied to the determination of stellar radial ve- 
locities. The student who wishes to pursue the subject further, 
will find a list of references to extended treatment of the 
various topics at the end of this chapter. 

Since stellar light sources are very faint in comparison with 
those available in the laboratory, it is necessary to employ for 
this special problem the spectroscope which is the least wasteful 
of light. For this reason the prism-spectroscope is the only 
one of the various laboratory forms which is at present gen- 
erally applicable to stellar spectroscopy and we, therefore, 
limit our discussion to this particular type. 

The essential parts of a laboratory spectroscope and their 
principal functions are briefly as follows: Light from the 
source to be studied is brought to a focus by a condensing lens 
on the narrow slit of the spectroscope. After passing through 
the slit, the rays are rendered parallel by an achromatic con- 
verging lens, called the 'collimator' lens. The rays then strike 
a glass prism, placed with its apex parallel to the length of 
the slit, by which they are bent from their original direction. 
It is here that we obtain the separation of the rays, since the 
amount by which each ray is deviated by the prism is a func- 
tion of its wave-length. The direction of the long red waves 
is changed the least, while the shorter violet ones suffer the 
greatest deviation. After each set of rays is collected and 
brought to its corresponding focus by a second achromatic 
converging lens, we shall have an orderly array of images of 
the slit, each image formed by light of a definite wave-length. 
Such a series of images is called a 'spectrum' of the source. 
The spectrum may be viewed with an ordinary eye-piece, or 
the second lens may be used as a camera lens, and the spec- 
trum be recorded on a photographic plate placed in its focal 
plane. In all stellar work the spectroscope is employed photo- 



no THE BINARY STARS 

graphically, in which case it is called a 'spectrograph', and the 
photograph obtained with it is a 'spectrogram'. 

If the slit is made extremely narrow there will be very little 
overlapping of the images and the spectrum is then said to be 
'pure'. It can be shown that the purest spectrum is obtained 
when the incident rays fall upon the prism at such an angle 
that they will be least deviated from their original direction 
by the prism. It is well known that this position of minimum 
deviation is also the one of maximum light transmission by the 
prism; and it has the further advantage that any accidental 
displacement of the prism produces the minimum displace- 
ment of the spectrum line. The prism or prisms of stellar 
spectrographs are therefore always set at the angle of mini- 
mum deviation for the approximate center of the region of 
spectrum to be studied. 

Attention was called in an earlier chapter to two factors 
which define the optical efficiency of a telescope for the sepa- 
ration of close double stars, viz: (a) the resolving power of 
the objective, (b) the magnification or linear distance between 
the two images at the focus of the objective. These same 
factors form a convenient basis for the comparison of the 
resolving powers of two spectrographs. Here, however, we 
are concerned with the separation of two images of the slit 
formed by light of different wave-lengths. The resolving 
power of a spectrograph is, therefore, defined as the minimum 
difference of wave-length between two lines for which the lines will 
just be separated. It is a function of the width of slit, the wave- 
length, and the difference between the maximum and minimum 
lengths of path of the rays in the prism. The magnification, 
called 'the linear dispersion' of the spectrograph, is expressed, 
as the number of wave-length units per unit length of spectrum 
and depends upon the wave-length of the ray, the optical con- 
stants for the prism system and the focal length of the camera 
lens. 

When the slit of a spectroscope is illuminated by the light 
from an incandescent solid, such as the filament of an incan- 
descent lamp, or from an incandescent gas under high pressure, 
the spectrum consists of an unbroken band of color; that is, a 



THE BINARY STARS III 

continuous spectrum. An incandescent gas or vapor under 
low pressure gives a spectrum consisting of isolated bright 
line images of the slit — a bright line spectrum — the bright 
lines indicating that radiations of certain definite wave-length 
are emitted by the gas. Each chemical element, in the gaseous 
state, when rendered luminous in the electric arc, electric 
spark, flame, or vacuum tube, gives its own set of bright lines, 
which are characteristic of this element alone and whose wave- 
lengths remain constant for a source at rest under the same 
conditions of temperature, pressure, etc. 

An incandescent gas has the property not only of radiating 
light of certain definite wave-lengths, but also of absorbing, 
from white light passing through it, the rays of precisely those 
same wave-lengths. If the temperature of the incandescent 
gas is lower than that of the source behind it, the continuous 
spectrum will be crossed by relatively dark lines, whose posi- 
tions agree exactly with the bright line spectrum characteristic 
of the gas. This relation existing between the emission and 
absorption of a gas is known as Kirchoff's law, and the type 
of spectrum described is termed an absorption spectrum. 

The three principles just stated obviously lead to a simple 
and direct method of analyzing the chemical constituents of 
gaseous light sources, and of furnishing information as to their 
physical conditions. A certain class of nebulae, for example, 
give bright line spectra, indicating that they are masses of 
luminous and extremely rarefied gases. Most of the stars, 
including our own sun, give absorption spectra, showing that 
the light emitted by a central glowing core has passed through 
a surrounding atmosphere of cooler vapors. The presence of 
most of the known chemical elements in the atmospheres of the 
Sun and stars has been recognized from the lines in the spectra 
of these objects. In addition, there occur in them many lines, 
which have not yet been identified with those of any known 
element. 

The length of the light wave for each line is such a minute 
fraction of a millimeter that spectroscopists have adopted as 
the unit of wave-length, the 'Angstrom', equal to one ten-mil- 
lionth of a millimeter, for which A is the symbol. Thus the 



112 THE BINARY STARS 

wave-length of the hydrogen radiation in the violet is 0.0004340 
mm. or 4340 A. 

Measures of the wave-lengths of the lines in a star's spec- 
trum are readily effected with the prism spectrograph, by a 
comparison of the positions of the stellar lines with those from 
a source the wave-lengths of whose lines are known. To 
accomplish this the light from a suitable source (for example 
the iron arc) is made to pass over very nearly the same path 
in the spectrograph as the star's light travels, and the spec- 
trum of this source, termed the comparison spectrum, is 
recorded on each side of the star spectrum. 

When the spectra of a number of stars are examined, it is 
found that they exhibit a great variety in the number and 
character of their lines. From an examination of several 
hundred stars by means of a visual spectroscope, Secchi about 
1 866-1 867 was able to arrange their spectra under four types. 
While exhibiting very well the most prominent characteristics 
of stellar spectra, his system is insufficient for portraying the 
finer gradations, which the photographic method has brought 
to light. The classification now in general use among astro- 
physicists, was formulated by Professor Pickering, Miss 
Maury, and Miss Cannon from the very extensive photo- 
graphic survey of stellar spectra made at the Harvard College 
Observatory and at the Harvard station at Arequipa, Peru. 
It is based upon the observed fact that certain groups of lines 
have a common behavior. They make their appearance and 
increase or decrease in intensity at the same time, so that a 
more or less orderly sequence of development from one type 
of spectrum to another is indicated. 

A very condensed outline of this system of classification will 
serve to indicate its chief features. Its main divisions, ar- 
ranged in the supposed order of development, which is that 
of the more generally accepted order of stellar evolution, are 
represented by the capital letters P, 0, B, A, F, G, K, M, 
{R, N). Sub-groups are indicated by small letters or 
numbers on the scale of ten. To class P are assigned all 
bright line nebulae, while the other classes refer to stellar 
spectra. 



THE BINARY STARS II3 

Spectra of Class 0, in the five sub-divisions Oa to Oe, con- 
tain a group of bright bands of unknown origin, and also the 
first and second series of hydrogen lines, which are bright in 
Oa-Oc, and dark in Od and Oe. Toward the end of the class 
some of the so-called 'Orion lines', or dark lines due chiefly to 
helium, nitrogen, silicon, magnesium and carbon, begin to 
appear. In Class Oe^, intermediate between Oe and B, the 
bright bands have disappeared. The secondary series of 
hydrogen vanishes early in Class B, while the primary series 
increases in intensity throughout the ten subdivisions. Bo, Bi, 
etc. Near the middle of the group the Orion lines begin to 
disappear, and toward the end, in B8 and Bq, some of the 
metallic lines are faintly visible. In the Classes Ao and A2, 
the primary hydrogen series reach their maximum intensity 
and decrease in the other two subdivisions, Aj and A^. The 
calcium lines, H and K, and those due to the metals increase 
in prominence through this class and the four subdivisions of 
Class F. In Class G, which includes stars whose spectra 
closely resemble that of the Sun, the H and K lines and a 
band designated by g are the most conspicuous features, 
whereas the hydrogen lines are scarcely more prominent than 
many of the metallic lines. Classes G5, Ko, and K2, represent 
spectra of a type a little more advanced than that of the Sun. 
Class K is further characterized by a decrease in intensity of 
the continuous spectrum in the violet and blue. This becomes 
quite marked in Classes K^ and Ma, Mb, and Md. The three 
divisions of Class M are further distinguished by absorption 
bands of titanium oxide, which first make their appearance 
in K^. Stars of Class Md show in addition bright hydrogen 
lines. To Class N belong stars whose spectrum of metallic 
lines is similar to that of M, but which are particularly char- 
acterized by a banded spectrum ascribed to carbon absorp- 
tion. Class R includes stars whose spectra are similar to those 
of Class N, except that they are relatively more intense in the 
violet. These two classes we have placed in brackets in the 
arrangement according to development, since some uncer- 
tainty exists as to the place they should occupy in such a 
scheme. Stars of Classes 0, B, and A are bluish white in color. 



114 THE BINARY STARS 

F, G and K stars are yellow. Those of Class M are red or 
orange, while the N stars are a deep red. 

In Plate III are reproduced four stellar spectrograms secured 
with the three-prism spectrograph of the D. O. Mills Expedi- 
tion, at Santiago, Chile, which illustrate the different appear- 
ance of the spectra in the blue-violet region of Classes B8, 
F, G, and K^. On all of the spectrograms the bright line 
spectrum of the iron arc was photographed above and below 
the star spectrum. The spectrum of U4 Eridani (Figure a) 
of Class B8, shows only the hydrogen line Hy (4340.634^) 
and the magnesium line (4481.400^), as the very faint metallic 
lines, some of which appear on the original negative, are lost 
in the process of reproduction. This star is a spectroscopic 
binary, and the spectra of both stars are visible, so that each 
of the two lines mentioned above is double. The strengthen- 
ing of the metallic lines and the decrease in intensity of H7 
is shown in the spectrum of a Carinae of Class F (Fig. 6), while 
in the solar spectrum (Fig. c), of Class G, and in that of a2 
Centauri (Fig. d), of Class K^, a further decrease in Hy, the 
disappearance of 448 i^l and a considerable increase in the 
number and strength of the absorption lines of other elements 
are noticeable. 

The four spectrograms illustrate also the displacement of 
the lines in star spectra as efifects of motion in the line of sight. 
The iron lines in the solar spectrum are practically coincident 
with the corresponding lines of the iron arc, since the relative 
radial velocity of the Sun and the observer is very small. The 
iron lines in the spectrum of a Carinae are clearly displaced 
from their normal positions, as given by the lines of the com- 
parison spectrum. This displacement is toward the red end 
of the spectrum, and corresponds, therefore, to an increase in 
the wave-lengths of the star lines. Interpreted on the Doppler- 
Fizeau principle, this change is produced by a recession of the 
star with respect to the Earth at the rate of +25.1 km. per 
second. In the case of a2 Centauri, the displacement of the 
lines is toward the violet and corresponds to a velocity of 
approach of —41.3 km. per second. As an example of the 
Doppler-Fizeau effect, the spectrogram of the spectroscopic 



THE BINARY STARS 115 

binary 1^4 Eridani, is perhaps the most striking. The two stars 
revolve about their common center of mass in a period of 5.01 
days, as shown by an extended series of plates similar to this 
one. Due to their orbital motion, the velocity of each star 
in the line of sight is continually changing, giving rise to a 
continuous variation in the separation of the lines of the two 
spectra. The spectrogram reproduced here, was taken at the 
time of maximum velocity of approach of one, and the cor- 
responding velocity of recession of the other component. It 
shows, therefore, the maximum separation of the lines of the 
two spectra. The relative radial velocity of the two stars was 
126 km. per second. Obviously, the lines of the two spectra 
will be coincident when the motion of the two components is 
across the line of sight, which occurs at intervals of 2.5 days. 

It is well known that the wave-lengths of spectral lines 
are affected by other causes than that arising from radial 
motion of the source. For example, it is found that an increase 
in pressure of the emitting or absorbing vapor will in general 
shift the lines toward the red. This effect, even with consider- 
able pressures, is small and is moreover not the same for all 
lines. Of the many conditions which displace spectrum lines, 
radial motion is the only one of which measures of stellar 
spectra have furnished reliable evidence. 

Displacements of the stellar lines with reference to those of 
the comparison spectrum, may arise wholly or in part from 
causes which are purely instrumental. Thus, if the star light 
and the artificial light do not pass over equivalent paths in 
the spectrograph, or if a change in the relative positions of the 
parts of the instrument occurs between the times of photo- 
graphing the stellar and the reference spectra, a relative dis- 
placement of the lines of the two spectra will result. The first- 
named source of error is an optical condition, to be met for 
all spectroscopic measures, that is easily satisfied. With the 
conditions of a fixed mounting and approximately constant 
temperature, under which the spectrograph is used in the 
laboratory, the second source of error need not be considered. 
When, however, the spectrograph is applied to stellar observa- 
tion, it is necessary, in order to avoid undue loss of light, to 



Il6 THE BINARY STARS 

mount it on a moving telescope, and hence to subject the in- 
strument to the varying component of gravity and the chang- 
ing temperature of a well-ventilated dome. The spectrograph 
must be so designed and constructed that it will be free from 
appreciable differential flexure in any two positions of the 
telescope, and provision must be made against the disturbing 
effects of temperature changes in the prisms and of the metal 
parts of the instrument. Further, in addition to the obvious 
requirement that the prisms and lenses shall give good defi- 
nition, they must be so chosen and arranged as to give satis- 
factory resolving power with efficiency in light transmission. 

The earlier determinations of stellar radial velocities were 
made entirely by the visual method. Although made by such 
skilled observers as Huggins, Vogel, and others, the errors of 
observation, except for a very few of the brightest stars, often 
exceeded the quantities to be measured. After the introduc- 
tion of the photographic method of studying stellar spectra, 
Vogel and Scheiner, at Potsdam, and later Belopolsky, at 
Pulkowa, were able to measure the radial velocities of the 
brightest stars with an average probable error of =±=2.6 km. 
per second. In 1 895-1 896 the problem was attacked by Camp- 
bell, who employed a specially designed stellar spectrograph — 
the Mills Spectrograph — in conjunction with the thirty-six- 
inch refractor of the Lick Observatory. For the brighter stars, 
the probable error of his measures was about =^=0.5 km. and 
for bright stars whose spectra contain the best lines, the 
probable error was reduced to =^=0.25 km. Many improve- 
ments in stellar spectrographs have, of course, been made in 
the succeeding twenty-one years, but the standard of preci- 
sion set by his measures represents that attained today for 
the same stars. The advances which have been made in 
this time relate more to the increased accuracy of the results 
for fainter stars. 

Now this remarkable advance in the precision of the meas- 
ures made by Campbell was due not to the use of a great tele- 
scope but to the fact that his spectrograph was designed in 
accordance with the important requirements mentioned above- 
— excellence of definition and maximum light transmission, 




Plate IV. The Mills Spectrograph of the Lick Observatory 



THE BINARY STARS II7 

rigidity and temperature control of the spectrograph — and to 
improved methods of measuring and reducing the spectro- 
grams. 

In order to understand more clearly the manner in which 
the optical and mechanical requirements are met in practice, 
a detailed description will be given of a modern spectrograph 
which was designed to have maximum efficiency for the par- 
ticular problem of determining stellar radial velocities. A 
view of the new Mills spectrograph attached to the thirty-six- 
inch refractor of the Lick Observatory is presented in Plate 
IV. The essential parts of this instrument are the same as 
those described for the simple laboratory spectrograph; 
namely, the slit, collimator lens, prism and camera lens, 
except that here three 60° prisms of flint glass are employed. 
The prisms, set at minimum deviation for 4500 A, produce a 
deviation of this ray of 176°. A rectangular box constructed 
of saw-steel plates, to which are connected respectively the 
slit mechanism, the prism box, and the plate holder, by three 
light steel castings, forms the main body of the spectrograph. 
In the casting to which the prism box is attached are mounted 
the collimator and camera lenses, both of which are achro- 
matic for the region of 4500 A. The spectrograph has an 
entirely new form of support, designed by Campbell, to in- 
corporate the suggestion made by Wright, that such an 
instrument should be supported near its two ends, like a biidge 
truss or beam, in order to give minimum flexure. The support 
is a frame work of T-bars extending down from the telescope, 
the form and arrangement of which is such as to hold the in- 
strument rigidly in the line of collimation of the large tele- 
scope. The lower support is a bar passing through a rec- 
tangular opening in the casting carrying the prism box. This 
bar is pivoted at the center of the casting and connected at 
its two ends to the supporting frame. The upper support 
consists of a cylindrical ring firmly attached to the frame 
work. In this cylinder fits a spherical flange of the spectro- 
graph casting, the two forming a universal joint. Any strains 
originating in the supporting frame cannot, with this form of 
mounting, be communicated to the spectrograph. Careful 



Il8 THE BINARY STARS 

tests of this instrument and of the spectrograph of the D. O. 
Mills Expedition to Chile, which has the same form of mount- 
ing, show that the effects of differential flexure have been 
eliminated. This method of support permits, further, of a 
very convenient mode of moving the spectrograph as a whole 
in order to bring the slit into the focal plane of the large tele- 
scope, since it is only necessary to provide sliding connections 
on the frame, for the lower support. 

Nearly all modern stellar spectrographs are provided with 
reflecting slit plates inclined at a small angle to the collimation 
axis, which enable the observer to view the star image directly 
on the slit. This is accomplished through the aid of a total- 
reflection prism, placed above the slit and outside of the cone 
of rays from the telescope objective, which receives the light 
from the slit and sends it to the guiding eyepiece. By placing 
the slit parallel to the celestial equator, small errors of the 
driving clock cause the star image to move along the slit, 
which is desirable in order to obtain width of spectrum. Con- 
stant and careful guiding is necessary to insure that the star's 
image he kept exactly on the slit and that its motion along the 
slit be such as to give a uniform exposure. 

With a prism spectrograph and a straight slit the spectrum 
lines are curved. The amount of the curvature depends upon 
the optical constants of the instrument and the wave-length 
of the line. This source of trouble in measuring the spectro- 
grams may be eliminated for a short range of spectrum by 
employing a slit of the proper curvatui^ to make the .spectrum 
lines straight. Both three-prism instruments referred to above 
are provided with curved slits. 

As a source for the comparison spectrum, it is necessary to 
select one giving a number of well-distributed lines in the part 
of the spectrum to be studied. For example, for the new 
Mills spectrograph in which the region 4400 A to 4600 A is 
utilized, the spark spectrum of titanium is used. In the 
southern instrument, arranged for the region 4200A-4500A, 
the comparison source is the iron arc. 

In order to eliminate the effects of any possible change in 
the instrument during an exposure on the star, several impres- 



THE BINARY STARS II9 

sions of the comparison spectrum are made at regular inter- 
vals. This is accomplished very conveniently and without 
danger of changing the adjustment of the comparison appara- 
tus by a simple device due to Wright. Two small total-reflec- 
tion prisms are placed just above the slit, so that their adjoin- 
ing edges define the length of the slit. Two light sources are 
then so arranged that the beam of each is brought to a focus 
on the slit by a small condensing lens after total reflection in 
its respective prism. 

The optical parts of the spectrograph should, of course, be 
mounted so that they cannot move, but care must be taken 
that they are not cramped. This caution is especially perti- 
nent with regard to the large prisms. In the Mills spectro- 
graphs the prisms rest upon hard-rubber blocks and are 
firmly clamped to one of the side plates of the prism box by 
light steel springs which press against their upper surface. 
Small hard-rubber stops prevent lateral motion of the prisms. 

In order to prevent the effects of changing temperature, the 
principal parts of the spectrograph are surrounded by a light 
wooden box, tined with felt. Over the felt surface are strung 
a number of turns of resistance wire. The regulation of the 
heating current is effected by means of a very sensitive mer- 
cury-in-glass thermostat by which the temperature inside of 
the prism box is held constant during the night's work to 
within a few hundredths of a degree Centigrade. 

The function of the telescope objective, for observations of 
stellar spectra, is that of a condensing lens and the brightness 
of the point image in the focal plane is directly proportional to 
the area of the lens and its transmission factor. If we had 
perfect 'seeing' we should receive in the slit of the spectro- 
graph, with the widths generally employed, about ninety per 
cent, of the light in the star image. Due to atmospheric dis- 
turbances the image of a star under average conditions of 
seeing, is a circular 'tremor* disc whose diameter is four or 
five times the width of the slit, so that the brightness of the 
spectrum is not proportional to the area of the objective but 
more nearly to its diameter. For example, the relative intensi- 
ties of stellar spectra obtained with the same spectrograph 



I20 THE BINARY STARS 

respectively upon the thirty-six-inch and twelve-inch refrac- 
tors of the Lick Observatory would be (allowing for the dif- 
ference of transmission of the two), about as two to one, since, 
for the photographic rays, the loss of light is for the former 
about fifty per cent, and for the latter about twenty-five per 
cent. When a visual refractor is used for spectroscopic work, 
it is necessary to render it achromatic for the photographic 
rays. This is accomplished for the thirty-six-inch refractor 
by a correcting lens of 2.5 inches aperture placed one meter 
inside the visual focus of the telescope. This lens introduces 
an additional loss of light of fully ten per cent. 

Since a silver-on-glass mirror has, under the best conditions, 
a high reflecting power, and since it is also free from chromatic 
aberration, it would seem that the reflector should be the more 
efficient telescope to use in connection with a stellar spectro- 
graph. The reflector, however, possesses its own disadvan- 
tages, one of which is that it is very sensitive to changes of 
temperature. Our experience with the thirty-six-inch refrac- 
tor at Mount Hamilton and the thirty-seven and one-half 
inch reflector in Chile, when used with high dispersion spec- 
trographs, indicates that the relative light efficiency of the 
two is about equal in the region of H7. For apertures up to 
thirty-six inches one is inclined to favor the refractor for high 
dispersion work, while for low dispersion, where considerable 
extent of spectrum is desired, the reflector is, of course, 
preferable. 

The focal lengths of both refracting and reflecting telescopes 
vary with change in temperature of the lens or mirror. It is, 
therefore, necessary before beginning the night's work, and 
with the reflector frequently during the night, to bring the 
slit into the focal plane of the telescope, which as noted above, 
is effected by moving the spectrograph as a whole in the line 
of collimation of the instrument. 

It is well known that all high dispersion spectrographs are 
very wasteful of light, though to what extent is perhaps not 
always appreciated. When stellar spectrographs of three- 
prism dispersion are used in conjunction with large refractors 
or reflectors the combined instrument delivers to the photo- 



THE BINARY STARS 121 

graphic plate probably less than two per cent, of the light 
incident upon the telescope objective. Half of the light is 
lost, as we have noted, before it reaches the slit. The remain- 
ing losses occur at the slit, in the prisms and in the collimator 
and camera lenses of the spectrograph. In order to avoid un- 
necessary losses of light, the obvious conditions must be 
satisfied, that the angular apertures of the collimator lens 
and object glass are the same, and that the prisms and camera 
lens are of sufficient aperture to admit the full beam from the 
collimator. The most serious losses occur at the narrow slit 
and in the prism train. Indeed, one of the most important 
factors in the design of stellar spectrographs, for maximum 
light efficiency, is the proper balancing of these two conflicting 
elements, the transmission at the slit and the transmission of 
the prisms. Thus, in the new Mills spectrograph, by using a 
collimator of slightly greater focal length than the present 
one (28.5 inches) with corresponding increase in aperture of 
the lens and prisms, a wider slit could be employed and still 
maintain the present purity of spectrum. After allowance 
is made for the increased absorption of the prism train, there 
would remain a small gain in light transmitted. Although 
theoretically possible, this gain would probably be more than 
offset by the inferior definition of the larger prisms and the 
added difficulty of eliminating flexure. It is necessary here, 
as at so many points in the spectrograph, to sacrifice a little 
in order to gain more elsewhere. In fact, the most efficient 
design of spectrograph may be described as the one in which 
the wisest compromises have been made between the various 
conflicting interests. 

The decision as to the resolution and dispersion to be em- 
ployed is governed by several considerations : the type of stellar 
spectrum to be studied, the size of the telescope at one's dis- 
posal, and the brightness of the source whose spectrum can 
be photographed with reasonable exposure times. With the 
spectrograph here described two lines in the region of 4500 A 
whose wave-lengths differ 0.2 A are resolved, while the linear 
dispersion for 4500 A is I mm. = 11 A. In order to obtain 
a spectrogram of suitable density of a star whose photographic 



122 THE BINARY STARS 

magnitude is 5.0, an exposure time of an hour and a half is 
required. For stars of photographic magnitudes 6.0 to 6.5 
the width of slit is increased, thus sacrificing to some degree 
the purity of spectrum, but not enough to interfere seriously 
with the accuracy of the measures. In the case of early type 
stars whose spectra contain single lines, the question of reso- 
lution is not important, and where these lines are also broad, 
it is preferable to employ lower dispersion. The adjustments 
of the various parts of the spectrograph call for continual 
attention. It is necessary that the instrument be placed with 
its axis of collimation accurately in that of the large telescope 
and frequent tests should be made to be sure that it remains 
so. The comparison source must be adjusted so that its light 
follows very nearly the same path as the star light in the spec- 
trograph. Care must be exercised at every point in the pro- 
cess of obtaining and measuring the spectrogram. 

THE MEASUREMENT AND REDUCTION OF SPECTROGRAMS 

For the measurement of spectrograms any one of the usual 
forms of laboratory measuring microscopes will suffice. This 
is merely a microscope on the stand of which is mounted a 
carriage, movable by an accurate micrometer screw, in a 
direction at right angles to that of the microscope axis. 

In order to fix ideas, we shall assume that it is required to 
measure and reduce the spectrogram of a2 Centauri, the posi- 
tive of which is reproduced in Plate III. The spectrogram is 
first clamped on the carriage of the microscope, and the usual 
adjustments of focus and alignment of the plate are made. 
Great care should be taken that the illumination of the field 
of the microscope is uniform. Beginning with the comparison 
line 4250 A, settings are made continuously along the plate on 
good star lines and comparison lines as they chance to occur. 
The plate is then reversed and the settings are repeated. It has 
been shown by several investigators that the effects of errors 
due to personal equation are practically eliminated by taking 
the mean of the measures in the two positions. In the reversal 
of the plate the spectrum is also inverted, which may so change 
the appearance of the lines as to interfere with the elimination 



THE BINARY STARS I23 

of personal equation. Especially is this true if the lines are 
curved. The effects of accidental errors in setting are reduced 
by employing a number of lines. 

The accompanying table contains the data of the measure 
and reduction of this plate. Column I gives the wave-lengths 
of the lines of the iron comparison and the normal wave- 
lengths of the star lines, taken from Rowland's 'Preliminary 
Table of Solar Wave-lengths'. In columns IV and V are 
recorded respectively the settings on the comparison and star 
lines (in revolutions of the micrometer screw). The displace- 
ments of the iron lines in the star are evidently given directly 
in amount and sign by the difference, star minus comparison 
and these are entered at once in column VII (Displ.). We 
cannot enter the displacements for the other star lines until 
the normal positions of these lines have been obtained from 
those of the iron comparison, by interpolation. This is effected 
in the following manner: 

A smooth curve drawn by plotting, for the comparison lines, 
the reading on each line and its corresponding wave-length, 
respectively as ordinates and abscissae, will evidently repre- 
sent for this spectrogram the relation existing between wave- 
length and micrometer readings. From this curve — called a 
'dispersion curve' — either the zero readings or the observed 
wave-length of the stellar lines could be obtained. This curve 
was found by Cornu and later by Hartmann to be nearly of 
the form of an equilateral hyperbola so that it is approximately 
represented by the equation 

X — Xo = T- , (2) 

A — Ao 

where x is the micrometer reading on a line whose wave-length 
is X and Xo, Xo, and c are constants. Since it is not practicable 
to plot » the dispersion curve, the Cornu-Hartmann formula 
furnishes a very convenient means of obtaining it. The values 
of the three constants are determined from three equations 
formed by substituting the micrometer readings and wave- 
lengths of three lines, selected, one at each end of the region 
of spectrum and the other near the middle. Micrometer read- 



124 



* 02 Centauri ft. 
Date 191 1 Feb. 27 



THE BINARY STARS 



♦Plate No. 3791 III a 14^ 32.8' 



X 


Table 


Co- 
Ta 


Comp. 


4> 


Stip'd D 


ispl 


rVs 


Vs 


4250.287 


54-886 





54-886 


54-758 


— . 


128 


319 


—40.8 


4250.945 


55031 


3 


55-034 


54-909 


— . 


125 


320 


— 40.0 


4282.565 


61.819 


13 


61.832 


61.710 


— . 


122 


335 


-40.9 


4283.169 


61 . 944 






61.831 


958 - 


127 


335 


-42.5 


4294.301 


64 - 250 


16 


64.266 


64. 140 


- 


126 


338 


— 42.6 


4299.410 


65-295 


20 


65-315 


65.190 


— 


125 


340 


-42.5 


4313- 034 


68.039 






67.944 


061 - 


117 


349 


-40.8 


4313-797 


68.190 






68.090 


212 - 


122 


349 


— 42.6 


4318.817 


69.185 






69.105 


220 — 


115 


352 


-40.5 


4325- 152 


70.431 






70-355 


469 - 


114 


356 


— 40.6 


4325- 939 


70.584 


40 


70.624 


70.502 


— 


122 


356 


-43.4 


4328.080 


71.001 






70.928 


041 — 


113 


357 


-40.3 


4337.216 


72.767 


43 


72.810 


72.692 


- 


118 


360 


-42.5 


4340.634 


73.421 






73.350 


467 - 


117 


362 


-42.4 


4359- 784 


77.027 






76.970 


082 - 


112 


372 


-41.7 


4369.941 


78.896 






78.844 


957 - 


113 


376 


-42.5 


4376.107 


80.018 






79.972 


083 - 


III 


378 


— 42.0 


4379 396 


80.612 






80.571 


680 - 


109 


380 


-41.4 


4383.720 


81.388 


70 


81.458 


81.352 


- 


106 


382 


-40.5 


4399 935 


84-257 






84.228 


337 - 


109 


390 


-42.5 


4404.927 


85.126 


86 


85.212 


85.105 


— 


107 


392 


-41.9 


4406.810 


85.453 






85.432 


539 - 


107 


394 


-42.2 


4415- 293 


86.913 


93 


87.006 


86.898 


— 


108 


397 


-42.9 


4425.608 


88.664 






88.662 


759 - 


097 


402 


-39.0 


4428.711 


89.198 






89.194 


296 - 


102 


404 


-41.2 


4430.785 


89-536 






89-535 


636 - 


lOI 


404 


-40.8 


4435- 129 


90.270 






90.270 


372 - 


102 


406 


-41.4 


4435.851 


90.380 






90.378 


482 - 


104 


406 


-42.2 


4442.510 


91 . 482 


108 


91-590 


91.491 


- 


099 


411 


-40.7 


4443.976 


91.724 






91.732 


831 - 


099 


412 


-40.8 


4447.892 


92.365 






92.375 


473 - 


098 


413 


-40.5 


4459-301 


94.216 


114 


94-330 


94-238 


— 


092 


417 


-38.4 


4476.185 


96.906 


127 


97 033 


96.940 


— 


093 


426 


-39.6 


4482 . 379 


97.872 


131 


98.003 


97 905 


- 


.098 


428 


-41.9 


4494- 738 


99.782 


138 


99.920 


99 . 820 


— 


.100 


434 


-43.4 



35)1449.9 
-4143 

Scale= +0.13 

va= +21.82 

vd= — 0.07 

Observed F —19.55 km. 



THE BINARY STARS I25 

ings of all other comparison and star lines are then computed 
from the formula. The departure of this computed curve from 
the true dispersion curve is furnished by a plot of the differ- 
ences between the observed and calculated readings of the 
comparison lines. The computed normal positions of the star 
lines are then corrected for the difference between the com- 
puted and observed dispersion curve. The decimal portions of 
the results would be entered in column VI (Sup'd).^ 

As before, the difference, star line minus zero line, gives the 
displacement in revolutions of the screw. In order to express 
this as AX, that is in units of wave-length, it is necessary to 
know r, the number of angstrom units in one revolution of the 
screw. The value of r for any point in the spectrum is evi- 
dently the slope of the dispersion curve at that point, and is 

equal to —- Finally, in accordance with the relation de- 

duced earlier, v the observed radial velocity is obtained by 
multiplying AX for each line by its corresponding factor 
_ 299860 

Each spectrogram may be reduced in the manner outlined 
above, and some observers prefer to follow this method rigor- 
ously for each stellar spectrogram. When this is done the 
process is simplified by carrying through the computation in 
wave-lengths, so that the displacement is expressed at once 
in angstroms. 

Since for the same spectrograph the form of the dispersion 
curve differs but slightly for different temperatures, a simple 
and practical method of reduction is offered by the following 
procedure: A standard dispersion curve is computed once for 
all, according to the method described above, from measures 
of a solar spectrogram. With the aid of this all other spectro- 
grams taken with the spectrograph may be quickly and easily 
reduced. It is convenient to put this standard curve in 
the form of a dispersion table in which are entered the 
normal wave-lengths of the comparison and stellar lines used 

2 The figures actually entered in this column in the example were obtained by a different 
method of reduction which is explained in the paragraphs following. 



126 THE BINARY STARS 

for stars of different spectral classes, and the micrometer 
readings corresponding to these wave-lengths. In this stan- 
dard table are given also the values of rVs for each line. 
Columns I and II and VIII, in the example, are taken from 
such a table. 

It is now only necessary to reduce the readings of the 
standard table to the dispersion of the plate, by plotting the 
differences between the observed and table readings; of the 
comparison lines (recorded in Column III in the example). 
From this curve the difference to be applied to the table read- 
ing for each star line is read off. In the sixth column are given 
the new table readings (for zero velocity) after this difference 
has been applied. When there are comparison lines cor- 
responding to star lines some observers follow rigorously the 
process outlined, while others (as in the present example) 
take the difference between the readings of the two as the 
displacements. The last three columns contain, respectively, 
the displacements (* minus Comp. or Sup'd), the factor rVs, 
and the products of these two values, which are the relative radial 
velocities of star and observer as supplied by the lines mea- 
sured. The mean of the measures for forty lines gives as the 
observed radial velocity —41.43 km. /sec. It will be noticed 
that the dispersion of the star plate is about three-tenths of one 
per cent, greater than that of the standard table, and conse- 
quently the factor r (computed for the table) is too large, and 
the numerical value of this velocity must be reduced by this 
amount. This is allowed for, in the example, as scale correc- 
tion. In practice, it is convenient to have several standard 
tables corresponding to the dispersion of the spectrograph at 
different temperatures. The one whose dispersion is nearest 
that of the star plate is selected for use. Experience has 
shown that the results obtained by the very simple method 
just described are of the same accuracy as those derived by 
the longer process of computing a dispersion curve for each 
plate. 

If the spectrograph is not provided with a curved slit it is 
necessary to introduce a correction for the curvature of the 
lines. This correction may be computed from Ditscheiner's 



THE BINARY STARS I27 

formula^ or determined empirically from lines on a spectrogram 
of the Sun, on the assumption that the curve of each line is a 
parabola. The better method is to eliminate the source of 
this correction by the use of a curved slit. 

The observed radial velocity of a star is made up of the star's 
velocity, V, with reference to the solar system, and the 
velocity of the observer in the solar system. The latter con- 
sists of three components, which arise from (one) the revolution 
of the Earth around the Sun ; (two) the rotation of the Earth 
on its axis; (three) the revolution of the Earth around the 
center of mass of the Earth-Moon system. This last compo- 
nent never exceeds ±0.014 km. /sec. and may be neglected. 
The correction for the annual and diurnal motions of the 
Earth are readily computed from the formulae given by 
Campbell in Frost-Scheiner's Astronomical Spectroscopy 
(pp- 338-345). The values for these in the example are given 
respectively under Va and Vd. Hence, the observed radial 
velocity of a2 Centauri with reference to the Sun on 191 1, 
February 27.883 (Greenwich Mean Time) was— 19.55 km. / sec. 

Methods of reduction which depend upon dispersion for- 
mulae require an accurate knowledge of the wave-lengths of 
the lines used in both the comparison and stellar spectra. 
Accurate values of the absolute wave-lengths are not required 
but their relative values must be well determined. For exam- 
ple, a relative error of ±0.01 A in the wave-length of any line 
would produce an error in the velocity for that line of nearly . 
a kilometer. Interferometer measures of the wave-lengths 
in the spectra of a number of elements are now available, but 
for the wave-lengths of solar lines it is still necessary to use 
the determinations by Rowland. It has been shown that 
errors exist in Rowland's tables, amounting in some cases to 
as much as o.oi or 0.02A. Another and much more serious 
difficulty arises, for stellar lines, from the fact that stellar 
spectrographs have not sufficient resolution to separate lines 
which were measured as separate lines by means of Rowland's 
more powerful instrument. It is the practice of many obser- 

• Cber die Kriimmung der Spectrallinien, Sitz. Ber. d. Math. Klasse d. k. Akad. Wien ' 
Bd. LI, Abth. II, 1865; also Frost-Scheiner, Astronomical Spectroscopy, p. 15, 1894. 



128 THE BINARY STARS 

vers, where two lines merge to form one line in the star spectrum, 
to take the mean of the wave-lengths of the component lines, 
weighted according to the intensities given by Rowland for 
those lines in the Sun. Wave-lengths based on estimates of 
intensity should naturally be regarded with suspicion, and in 
fact we do not know, until the entire plate has been reduced, 
whether we have chosen an erroneous wave-length or not. 
It is well known that various stellar lines and blends behave 
differently for stars of different types. The lines in solar type 
stars are assumed to have the same wave-lengths as similar 
lines in the Sun. In the case of stars of other spectral classes, 
the solar lines which occur can be used in determining the 
wave-lengths of the non-solar lines and blends. In this 
manner special tables are constructed for stars of different 
types. 

When spectrographs of lower dispersion and resolution than 
that of three prisms are employed for the measure of solar and 
later-type spectra, the effect of uncertainties in wave-length 
of the stellar lines, due to blends, becomes very serious. The 
two methods of measurement and reduction which follow 
eliminate the sources of error incident to the use of blends, and 
erroneous wave-lengths as far as it is possible to do so. The 
first is that due to Professor R. H. Curtiss and is called by him 
the 'Velocity Standard Method'. In principle it amounts to a 
determination of the wave-lengths of the lines in the spectrum 
of a source whose radial velocity is known with the particular 
spectrograph which is to be used for measures of stellar spectra 
of this same class. Thus for the measures of spectra of the 
solar type, a table similar to the one we have described above 
is formed. The micrometer readings in this table, however, 
are not computed from assumed wave-lengths, but are the 
mean of the actual settings, on comparison and solar lines, 
obtained on several spectrograms of the Sun. These standard 
plates are produced as nearly as possible under the same con- 
ditions as the stellar plates to be measured. The procedure in 
the reduction of the measures by means of this table is then 
the same as that described above. It is necessary, of course, 
to correct the measured stellar velocity for the radial velocity 



THE BINARY STARS 129 

of the source when the standard spectrograms were taken. 
Standard tables for the reduction of measures of stars of other 
spectral classes may be formed in a similar manner, using as 
the standard sources stars whose radial velocities are well 
determined. 

The second method is due to Professor Hartmann, and is in 
principle the same as the preceding one, except that the star 
plate is referred directly to the standard plate on a special 
measuring microscope, known as the spectro-comparator. 
The instrument is provided with two plate carriages, one of 
which is movable. On one of the carriages the star plate is 
placed and on the other, which is provided with a fine microm- 
eter screw, is a standard plate of the Sun (taken with the 
stellar spectrograph). The microscope has two objectives so 
arranged that the images of portions of the two plates are 
brought, by means of total reflection prisms and a reflecting 
surface, to focus in the same plane and in the field of one eye- 
piece. By means of a silvered strip on the surface of one 
prism, the central portion of the Sun's spectrum is cut out and 
the star spectrum thrown into its place. In a similar manner, 
central strips of the comparison spectra of the Sun plate are 
replaced by those of the comparison spectra of the star plate. 
An ingenious arrangement of the microscopes permits of 
equalizing the scale of the two plates, by changing the relative 
magnifying powers of the two objectives. The method of 
measurement is, then, after proper alignment of the plates, 
to bring corresponding sections of the two plates into the field 
of the miscroscope, and by means of the micrometer screw 
set the corresponding lines of the comparison spectra in the 
same straight line. A setting is then made with the correspond- 
ing lines of the solar and star spectra in the same straight line. 
The difference between the micrometer readings in the two 
positions is the displacement of the star lines relative to the 
solar lines. In practice it is found sufficient to divide the 
length of the spectrum into about fifteen sections, for each 
of which these comparative settings are made. The mean of 
the displacements, obtained with the plates in the direct and 
reverse positions, when multiplied by the rVs for each section, 



130 THE BINARY STARS 

gives for each the value F* — Vq, where V* is the radial 
velocity of the star and Vq that of the Sun. Theoretically, 
the values of V* — Vq should receive weights proportional to 

——in taking the mean. Although this correction is negligi- 
ble, except where an extent of spectrum of 400 or 500 A is 
used, its introduction leads to a very simple method of com- 
putation. Take the sum of the displacements in the direct 

and reverse measures and multiply by a factor /= —. The 

product is equal to the weighted mean of the values F*— Vq 
for each section. This, corrected for the velocity of the original 
Sun plate {Vq), gives the radial velocity of the star relative 
to the observer. The reduction to the Sun is made in the 
usual way. The factor / is a constant so long as the same 
regions are used, and its values may be computed for all com- 
binations of the regions that are used. The great advantage 
of the method, aside from those which it possesses in common 
with the velocity standard method, is that we are able to 
measure and reduce in an hour a plate of a star rich in lines, 
and practically utilize all the material on the plate. With the 
older methods, to make such a comprehensive measure and 
reduction, i.e., to utilize all of the lines on the plate, would 
require one or two days. 

For the measures of spectra of a type other than the solar 
it is necessary to select for the standard plate a spectrogram 
of a star of that particular spectral class. In order to obtain 
the velocity for this standard spectrogram, it should be mea- 
sured and reduced, either by the method first described or per- 
haps preferably by the velocity-standard method. The 
adopted value should be the mean of the measures made by 
several different observers. 

The spectro-comparator offers a very efficient method in 
determining the differences in velocities of the same star, by 
measuring a series of plates of the star with reference to one 
of these selected as a standard. 

Five of the six elements of a spectroscopic binary orbit 
depend only upon the accurate determination of the relative 



THE BINARY STARS I3I 

radial velocities given by the series of spectrograms. One of 
the most important applications of the Hartmann comparator 
is, therefore, to the measurement of plates of a spectroscopic 
binary. 

For the measure and reduction of spectrograms of stars of 
the earlier spectral classes, the use of the Cornu-Hartmann dis- 
persion formula will suffice, inasmuch as the spectra of such 
stars consist of lines due to the simple gases, the wave-lengths 
of which have been accurately determined in the laboratory. 

The measure and reduction of spectrograms of stars of the 
solar and later classes of spectra are accomplished with great 
saving of time and labor, and by a method free from some of 
the uncertainties of wave-lengths, by the use of the spectro- 
comparator. If the observer is not provided with such an 
instrument the standard-velocity method is preferable to the 
use of the dispersion formulae, at least until a system of 
stellar wave-lengths of the requisite accuracy is available. 

To the reader who has followed the long and intricate process 
of determining the radial velocity of a star, the question will 
naturally occur, how do we know that the final result repre- 
sents the star's velocity? Obviously, the final test of the 
method is its ability to reproduce known velocities. Fortu- 
nately, we have at hand a means of making such a test. Since 
the orbital elements of the inner planets of the solar system 
are well determined, we can readily compute the radial 
velocity of one of these with reference to the Earth at any 
given time. It is only necessary, then, to observe the relative 
radial velocity of the planet and the Earth and compare this 
with the computed value at the time of observation. At the 
Lick Observatory spectrograms of Venus and of Mars are 
secured at frequent intervals with the stellar spectrograph 
and measured by the observers in the regular course of mea- 
suring stellar plates. With the three-prism spectrograph, de- 
scribed above, the observed and computed velocities of these 
two planets generally agree to within =1=0.5 km., or the una- 
voidable error of measure. When the spectrograms are 
measured by several observers, the effects of personal equa- 
tion are to some extent eliminated in the mean, and an agree- 



132 THE BINARY STARS 

ment within a few tenths of a kilometer is to be expected. A 
continual check is thus afforded on the adjustments of the 
spectrograph and the measurement of the spectrograms. 



REFERENCES 

General 
Campbell. Stellar Motions, Yale University Press, 1913. 

Doppler's Principle 

Kayser. Ilandbuch der Spectroskopie, Bd. 2. 

Frost-Scheiner. Astronomical Spectroscopy, chapter 2, part II. 

Instruments and Design 

Campbell. The Mills Spectrograph. Astrophysical Journal, vol. 8, 
p. 123, 1898. 

Frost. The Bruce Spectrograph. Astrophysical Journal, vol. i^, p. i, 
1902. 

Hartmann. Remarks on the Construction and Adjustment of Spectro- 
graphs. Astrophysical Journal, vol. 11, p. 400, 1900; and vol. 12, 
p. 31, 1900. 

Keeler. Elcmentar\' Principles Governing the Efficiency of Spectro- 
graphs for Astronomical Purposes. Sidereal Messenger, vol. 10, 
p. 433. 1891. 

Newall. On the General Design of Spectrographs to be Attached to 
Equatorials of Large Aperture, Considered Chiefly from the Point 
of View of Tremor-discs. Monthly Notices, Royal Astronomical 
Society, vol. 65, p. 608, 1905. 

VOGEL. Description of the Spectrographs for the Great Refractor at 
Potsdam. Astrophysical Journal, vol. 11, p. 393, 1900. 

Wright. Description of the Instruments and Methods of the D. O. 
Mills Expedition. Publications of the Lick Observatory, vol. g, 
part 3, 25, 1905. 

Methods of Measurement and Reduction 

Campbell. The Reduction of Spectroscopic Observations of Motions 
in the Line of Sight. Astronomy and Astrophysics, vol. 11, p. 319, 
1892. Also Frost-Scheiner, Astronomical Spectroscopy, p. 338. 



THE BINARY STARS 133 

CuRTiss. A Proposed Method for the Measurement and Reduction of 
Spectrogram for the Determination of the Radial Velocities of 
Celestial Objects. Lick Observatory Bulletin, vol. j, p. 19, 1904; 
Astrophysical Journal, vol. 20, p. 149, 1904. 

Hartmann. tlber die Ausmessung und Reduction der Photographi- 
schen Aufnahmen von Sternspectren. Astronomische Nachrichten, 
vol. 755, p. 81, 1901. 

. A simple Interpolation Formula for the Prismatic Spectrum. 

Astrophysical Journal, vol. 8, p. 218, 1898. 

-. The Spectro-comparator. Astrophysical Journal, vol. 24, p. 



285, 1906. Publikationen des Astrophysikalischen Observatorium 
zu Potsdam, vol. 18, p. 5, 1908. 



CHAPTER VI 

THE ORBIT OF A SPECTROSCOPIC BINARY STAR 

The problem of determining the orbit of a binary system 
from measures of radial velocity, made in the manner described 
in the previous chapter, differs in several important particulars 
from that of computing an orbit from micrometric measures of 
position angle and distance. It has been shown that microm- 
eter measures provide the data from which the projection of 
the orbit of the companion star with respect to its primary can 
be drawn, the true relative orbit following, correct in propor- 
tions but of unknown linear dimensions. The radial velocities, 
on the other hand, when plotted against the times, produce a 
periodic curve, having the general appearance of a distorted 
sine-curve; from this curve we are to find the elements of the 
true orbit of the star with respect to the center of gravity of 
the system of which it forms one component.^ 

Figure 7 illustrates the conditions of the problem. Let the 
XF-plane be tangent to the celestial sphere at the center of 
motion, and let the Z-axis, perpendicular to the JsTF-plane, be 
parallel to the line of sight along which the radial velocities 
are measured. The velocities are considered positive (+) when 
the star is receding from, and negative ( — ) when it is approaching 
the observer. The orientation of the X and F-axes remains 
unknown. Let PSA be the true orbit of the star with respect 
to the center of motion and let the orbit plane intersect the 
ZF-plane in the line NN'. 

Then, when the star is at any point S in its orbit, its distance 
z from the X F-plane will be 

z = rsmisin{v + co) 

1 It is here assumed that the spectrum of only one component is visible; when both com- 
ponents give spectra, we may determine the relative orbit of one with respect to the other, 
using the same formulae but changing the value of the constant of attraction. The relative 
and absolute orbits are, of course, similar in every respect. 



THE BINARY STARS 



135 



the symbols in the right hand member of the equation having 
the same significance as in the case of a visual binary star. 




Figure 7 

The spectrograph, however, does not give us the distances of 
the star from the XF-plane, but the velocities of its approach 
to, or recession from this plane, generally expressed in kilo- 
meters per second. The radial velocity at point S is equal to 
dzjdt, and is therefore expressed by 

dz ...... dr ^ . . dv 

— = sin t sin {v -f- CO) 1- ^ sin ^ cosCij + co) 

dt dt dt 

From the known laws of motion in an ellipse we have 
dv tia{i -\- ecosv) dr fxaesinv 



dt V 

and therefore 
ii a sin i 



dt 



Vi 



dz 
dt 



V] 



[ecosco + COS (d + 0))] (i) 



136 THE BINARY STARS 

which is the fundamental equation connecting the radial veloci- 
ties with the elements of the orbit.^ 

The observed velocities evidently contain the velocity, V, 
of the center of mass of the system, which is a constant quan- 
tity for any given simple binary system,^ as well as the variable 
velocities due to the star's orbital motion and the quantity V 
must therefore be subtracted from the observed values to make 
them purely periodic. In other words, the velocity curve is 
purely periodic only with respect to a line representing the 
velocity of the system as a whole. This line is called the 
F-axis. 

Equation (i) applies only to the velocities counted from the 
F-axis. If d^ /dt represents the velocity as actually observed 
(i.e., the velocity referred to the zero-axis) we shall have the 
relation ^ 

dt dt 

Methods of determining the position of the F-axis will be 
given later; for the present we shall assume it to be known. 

Five constants enter the right hand member of equation (i), 
viz., a sin i, e, n, co and (through v) T. These express the five 
orbit elements which it is possible to determine by measures 
of radial velocity. 

* In place of (» + co) the symbol m ( = the argument of the latitude) is often used, the 

expressions for — and r — written 

dt dt _ 

dr f . , . , du f\/p / r . ■ /■ \n 

— =x — ^ e sin (u — (a), and r — = ■'..u:— = — __ [i -\- ecos(u — w) J 

dt Vp dt r Vp 

and hence the fundamental equation in the form 

<*« / . w 

— = — =: sin » (cos u + e cos 00). 

dt Vp 

In these equations /> f = a (i — e)] is the semi-parameter of the true ellipse and / 
denotes the constant of attraction, which, when the spectrum of only one component is 
visible, and the motion is determined with reference to the center of mass of the system, 

takes the form , k being the Gaussian constant; when both spectra are visible and 

m + mi 

the motion of one star with respect to the other is determined, f = k v^m + mu It is clear 
that the form of the fundamental equation will be the same whatever value we may assign 
to'/ and the constant of attraction may therefore be disregarded until the question of the 
relative masses in the system comes up for discussion. 

3 In a triple or multiple system, this quantity will itself be variable. 

* The symbol 7 is often used for the velocity of the system instead of V. 



THE BINARY STARS 137 

Since the inclination of the orbit plane is not determinable, 
the value of a, the semi-major axis, must also remain unknown. 
It is therefore customary to regard the function a sin i as an 
element. Further, it is clear that the position of the line of 
nodes cannot be determined though we can find the times when 
the star passes through each of the nodal points. The various 
elements have the same definitions as in the case of visual 
binary star orbits (see page 72) except that the angle co in spec- 
troscopic binary orbits is always measured from the ascending 
node, the node at which the star is moving away from the observer. 
It will be seen later that the radial velocity has its maximum 
positive value at this node and its minimum positive value (or 
maximum negative value) at the descending node. It should 
also be noted that the unit of time for // (and therefore for P) 
is the day, not the year as in visual binary orbits. 

Theoretically, values of the radial velocity at five different 
times suffice for the complete solution of equation (i); prac- 
tically, no computer undertakes an orbit until a considerable 
number of measures is available which give the velocities at 
short intervals throughout the entire revolution period. To 
secure a satisfactory distribution of the observations a pre- 
liminary value of the period is necessary and such a value can 
ordinarily be obtained without difficulty by plotting the early 
observations on coordinate paper, taking the times, expressed 
in Julian days and decimals of a day, as abscissae and the 
velocities, expressed in kilometers per second, as ordinates. A 
convenient epoch as origin for the period is selected near the 
beginning of the series, preferably one corresponding to a 
point of maximum or minimum velocity. If later measures 
indicate that the period is in error, a new period which is a 
submultiple of the original one will often prove satisfactory. 
In difficult cases, the following artifice may be found helpful.'^ 
Copy from one-third to one-half of the series of observed 
points, choosing the time interval best covered by observa- 

* This was suggested to me by Dr. R. K. Young who says that it has been used with 
good results by several computers of binary star orbits at the Dominion Observatory. No 
mention of the device has been found in print and its author is unknown to me. Its use- 
fulness arises from the fact that, in effect, it doubles the number of observations for a given 
time interval. 



138 THE BINARY STARS 

tion, on transparent paper; slide the copy along the original 
plot, keeping the time-axis in coincidence, until some point 
on the copy falls approximately upon a different point in the 
original at which the velocity is changing in the same direction. 
The time interval between the two points is evidently equal 
to the period or a multiple of the period. 

Schlesinger ^ has published a criterion that may be applied 
to advantage in cases where an observer has accumulated 
many plates of a star which apparently shows variable radial 
velocity without being able to determine any period. It con- 
sists in constructing a frequency curve for the velocities by 
"dividing the total range exhibited by the measured velocities 
into successive groups of equal extent, say three kilometers 
each, and then counting the number of velocities that fall 
within these groups. Regarding these numbers as ordinates, 
we plot them and join the ends by a smooth curve." This 
curve is compared with the well-known error-curve; if the 
two are the same, within reasonable limits, we may conclude 
that the differences in the measured velocities are due to 
errors of observation, and afford no support for the assumption 
that the star is a spectroscopic binary. If the two curves 
differ, the star is a binary and the form of the frequency curve 
will give an idea as to the general character of the orbit and 
frequently furnish a clew to the period. For Schlesinger shows 
that circular orbits, elliptic orbits with periastron at descending 
node, elliptic orbits with periastron at ascending node, and 
elliptic orbits with periastron removed 90° from the nodes, all 
have characteristic frequency curves which differ in form from 
the error-curve. When the nature of the frequency curve has 
shown to which of these classes the orbit in question belongs, 
it becomes very much easier to decide upon the epochs for the 
various observed velocities, and thus upon an approximate 
value for the period. 

When the period is approximately known all of the observa- 
tions may be reduced to a single revolution by subtracting 

^ Astrophysical Journal, vol. 41, p. 162, 1915. In his paper on the "Orbit of the Spec- 
troscopic Binary X Aurigae" (Journal R. A. S. C. vol. X, p. 358), Young shows that the 
errors of measurement affect the expected distribution in such a manner as to mask to a 
considerable degree the presence of the orbital variation. 



THE BINARY STARS 139 

multiples of the period from the later dates. A preliminary 
curve is drawn to represent the plotted positions as closely as 
possible. The deviations from the curve at points near the 
mean of the maximum and minimum velocities, where a change 
in the periodic time will have the greatest effect, will indicate 
advisable changes in the assumed period and these are readily 
found by dividing the deviations of such critical observations, 
expressed in time, by the number of revolutions elapsed. A 
second curve is then drawn whose periodic time will generally 
be very close to the true value. In practice it will frequently 
happen that two or three measures of the radial velocity of a 
star are available which were made (perhaps at another obser- 
vatory) several years before the series of spectrograms for the 
orbit computation is begun. When an approximate value of 
the period has been found from the later series, these early 
plates will determine its true value with high precision. Gen- 
erally they are not used in finding the other orbit elements. 

When the period has been determined as accurately as 
possible and a series of spectrograms has been accumulated 
giving the velocities at points well distributed throughout the 
entire period, the most probable curve is drawn, by estimation, 
through the points as plotted, and, if the ingenious methods of 
superposition devised by Schwarzschild and Zurhellen are to be 
used, the curve should be prolonged through a revolution and 
a half. The plotted points used for this curve should repre- 
sent normal positions, formed by combining several velocities 
observed at very nearly the same orbit phase, whenever the 
number of observations is sufficient to permit such combina- 
tions. In making the combinations, the question of weights 
arises, and here the practice of computers varies considerably, 
for several factors enter. The character of the lines on the 
spectrograms, broad or narrow, sharp or ill-defined, strong or 
weak, is one factor; the number of lines is another; if the 
plates have been taken with different telescopes and spectro- 
graphs, a third factor is introduced. These must all be con- 
sidered in assigning the weights to each plate. The only direc- 
tion that can be given is the general one to use rather a simple 
system of weighting. It will rarely be of advantage to assign 



140 



THE BINARY STARS 



fractional weights, or to use a range of weights greater than, 
say, four units. The weights should, of course, be assigned to 
each plate, at the time of measurement. 

The errors in drawing the most probable curve have con- 
siderable effect upon the accuracy of the determination of the 
elements. At best the curve is not likely to be a perfect repre- 
sentation of the elliptic motion which caused it since it is 
natural to bend the curve slightly in or out at different points 
to satisfy the more or less exact observations. This difficulty 
is inherent and for it there is apparently no remedy other than 
that of testing the first orbit by a trial ephemeris and making 
the small changes in the elements which are indicated by the 
residuals.^ 

If Figure 8 represents a velocity curve, it is evident from 
equation (i) that the points A and B correspond respectively 
to the ascending and descending nodes of the star's orbit, for 
at the times of nodal passage we have (z; -f co) = 0°, and 
(y + 0)) = 180°, respectively and therefore cos (z; + co) = =»= i. 
The radial velocity thus reaches its maximum and minimum 
values at the nodal points. 

Taking A and B as the magnitudes of the curve-ordinates 
at the points of maximum and minimum reckoned from the 
F-axis, regarding 5 as a positive quantity and writing for 
IJL a sin i 



brevity K 



V, 



we have 



A =K{i 


-\- e cos co) 


B = K{i 


— e cos co) 


and therefore 


A-{-B 
2 


K 


A-B 
2 


K e cos CO 


A-B 


e cos CO 



A+B 



(2) 



"> King's method affords a graphical test of the first orbit found, see page 154. 



THE BINARY STARS 



141 




o lu 
2 > 



142 THE BINARY STARS 

Hence we may write equation (i) in the form 

— =K[e cos 0) + cos(i' + co) J = 1 cos {v -\- co) (3) 

at 22 

and (la) in the form 

F H 1 cos (y + co) = FiH cos (z^+cu). (3a) 

ft/ 2 2 2 

i<L is therefore the half-amplitude of the velocity curve. 

Up to the point now reached practically all methods of spec- 
troscopic orbit determination are identical. But when the 
fundamental relations are given as above, and the curve has 
been drawn, quite a variety of methods is available for com- 
puting the orbit elements, other than the period, which is 
assumed to be known. 

Of these, the method devised by Lehmann-Filhes will first 
be presented, essentially in full; other methods will then be 
treated in less detail. The student who desires to study the 
various methods more fully is referred to the important papers 
given in the references at the end of the chapter. 

METHOD OF LEHMANN-FILHES 

Given the observations, and the velocity curve drawn with 
the value of P assumed as known, the first step is to fix the 
F-axis, the line defining the velocity of the center of gravity 
of the system. This is found by the condition that the integral 
of dz jdt, that is, the area of the velocity curve, must be equal 
for the portions of the curve above and below the F-axis. By 
far the easiest method of performing this integration is to use 
a planimeter. A fine, approximately correct, is first drawn by 
estimation; the areas contained between it and the curve 
above and below are measured, and the difference between 
the two is taken. The position of the axis is then shifted to 
eliminate this difference, and the measures are repeated. It 
will rarely be necessary to make more than one correction to 
secure an accurate value of the position of the F-axis, which, 
by this method, depends upon the entire curve. 

If a planimeter is not available, the areas above and below 
the axis may be equalized by using coordinate paper for the 



THE BINARY STARS 143 

plot of the curve and counting the small squares in each area. 

Approximate mechanical integration, as advised by Lehmann- 

Filhes, may also be resorted to, by those who enjoy this form 

of recreation. 

Having found the F-axis, the ordinates to it are next drawn 

from the points of maximum and minimum velocity, A and B. 

It is at this point, as Curtis says, that the method is weakest, 

for slight errors in fixing the position of A and B may easily 

arise. It is well to apply the check afforded by the requirement 

that area AaC (Figure 8) must equal CbB and DaA equal BbD. 

Since C and D lie on the F-axis the velocities at these points 

dz 
are zero, hence from equations (3) and (2) we have for -— 

at 

at these points 

COS (v + o)) = = — e cos CO (4) 

A-{-B 

If Vi is the true anomaly corresponding to the point C, which 
is traversed by the star on the way from the ascending to the 
descending node, and V2y the true anomaly for the point D, 
sin (vi + co) will be positive, sin (v2 + co) negative, and we 
shall have 



cos {Vi + Co) = — -— , COS {V2 + CO) = - 

A + B A-\-B 

,n(., + co)= ___,sm(.. + co) = -^-p^ 



(5) 



Let Zi and Z2 denote the areas ^ AaC and bBD (Figure 8) 
respectively, and let ri and ^2 be the radii vectores for the 
points C and D. 

Then 

Zi = ri sin i sin {vi + co) 

Z2 = r2 sin i sin (1^2 + co) = — r2 sin i sin (f 1 + o)) 

and therefore 

Zi n i+ecost;2 ,^. 

(6) 



Z2 r2 i+^cosi'i 

8 These areas represent the distances of the star from the XF-plane at the points in its 
orbit corresponding to {vi + co) and (wj + co). 



144 THE BINARY STARS 

since r = . Write (v -\- co — co) for v, in (6), expand, 

i + ecost; 

and reduce, with the aid of the relations in (5) and (4), and 

we have 

Zi sin {vi + co) — e sin CO 



Z2 sin (t^i -f- w) + e sin CO 

whence 

Z2+Z1 . , , , 2^/AB Z2-\-Zi . . 

e sin CO = sin {vi + co) = . (7) 

Z2-Z1 ^ ^ ^ A +B Z2-Z, '^' 

Equations (7) and the last of (2) determine e and co. The 
values of A and B are taken from the curve, and the areas Zi 
and Z2 are quickly integrated from the curve portions AaC 
and bBD by means of a planimeter, the latter area being 
regarded as negative in sign. Since the areas enter as a ratio, 
the unit of area used is entirely immaterial. 

At the time of periastron passage v = o°\ hence from equa- 
tion (3) we have 

— =X(i 4-e)cosco (8) 
dtp 

which gives the ordinate corresponding to the point of peri- 
astron passage. Two points of the curve will have the same 
ordinate, but since (v + co) equals 0°, 180°, and 360° for the 
points A, B, and Ai, respectively, there will be no ambiguity 
as to the position of the periastron point. The abscissa of 
this point, properly combined with the epoch chosen for the 
beginning of the curve, defines T, the time of periastron pas- 
sage. Instead of using (8) we may find T by determining E 
for the point C for which the value of v is known, and then 
employ the formulae 



\i + l 



^ tan K V 



t- 



e 
E — esmE 



(8a) 



or, if the eccentricity is less than 0.77, M may be taken directly 
from the Allegheny Tables, and T found from the relation 

M = yL{t-T). (8b) 



THE BINARY STARS I45 

Such procedure is especially advisable when the periastron 
point falls near points ^4 or 5 on the curve. 

By definition (page 140) we have 

fjL a sin i 
K. = 



Vi- 

and hence 



AVi-e2 A-\-B Vi - 



a Sim 2 a sin t 



from which we may find the value of the product a sin i. 
Since the unit of time for A and B is the second, while for /x 
it is the day, the factor 86,400 must be introduced. Our 
equation then becomes 



K 



a sin i = 86,400 — V i -V = [4.i3833]ii:P Vi - e^ (9) 

the number in brackets being the logarithm of the quotient 
86400 -^ 2 TT. 

Summarizing, the practical procedure is: 

1. Find the period as accurately as possible by successive trials and 
plot the most probable velocity curve on the basis of normal places. 

2. Find the position of the F-axis by integration of areas, using the 
planimeter, if available. 

3. Measure the ordinates for points A and B and find the areas of 
AaC and bBD expressed in any convenient units. 

4. From (2) and (7) determine K, e, and w. 

5. From (8), or by calculation from the value of v, for the point C, 
determine T. 

6. From (9) determine a sin i. 

To test the elements by comparison with the observations, 
we compute the radial velocity for each date by the formulae : 

M = flit - T) = E - esinE 



-Vi? 



tan J/^ V =./-;—! — tan ^2 E 



— = V -\- Ke cos CO -\- K cos {v -\- co) 
dt 



(10) 



146 



THE BINARY STARS 



The value of v for each value of M may be taken directly from 
the Allegheny Tables, if e is less than 0.77. 

To illustrate Lehmann-Filhes's method I have chosen the 
orbit computed for k Velorum, by H. D. Curtis, the velocity 
curve for which is given in Figure 8. 

The observations used were as follows: 



Julian Day. G.M.T. 


Vel. 


Julian Day, G. M. T. 


Vel. 


2416546.739 


+ 68.5km 


2417686.591 


+ 33.8kni 


60.703 


+ 12.9 


91.572 


+ 38.2 


97651 


+ 65.7 


92.545 


+ 43-2 


6912.601 


+ 53.3 


96.480 


+ 46.7 


7587.844 


+ 58.6 


7701.494 


+ 52.7 


88.788 


+ 57.9 


41.466 


+ 22.1 


90.829 


+ 58.5 


46.463 


+ 0.3 


91.824 


+ 64.8 


49.470 


- 7.6 


97.788 


+ 65.8 


50.479 


- 8.8 


7609.790 


4- 62.0 


51.463 


- 13.3 


54.534 


— 21.0 


53.457 


- 19.2 


55.556 


- 19.2 


58.451 


— 29.0 


58.570 


- 15.2 


59.460 


— 24.6 


59.545 


- 14.5 







The small circles representing the first four observations, 
which are important in determining the period, owing to their 
distance in time from the later ones, are barred in the diagram. 
The period, P, was assumed to be 116.65 days, and the begin- 
ning of the curve is at Julian day 2416476.0. This is not 
exactly at a minimum, as may be seen from the diagram. 

From measures of the curve we find 

A = 46.3 Zi = AaC = H- 0.168 

B = 46.9 Z2 = bBD = - 0.259 

A -]- B = 93.2 Z2 + Zi = — 0.091 

A — B = — 0.6 Z2 — Zi = — 0.427 
K = {A-\- B)l2 = 46.6 



THE BINARY STARS 



147 



The solution of equations (2), (7), (8) then proceeds as 
follows : 



log 



log 2 


0.3010 


log (I + e) 


0.0828 


log Vab 


1.6684 


/oj COS CO 


8.48oon 


colog {A + B) 


8.0306 


logic 


1.6684 


log (Z2 + Zi) 


8.956on 
o.3696n 


\^^i ^' 


A o.23i2n 


colog (Z2 — Zi) 


^^Q^dtp 


/ 


log e sin co 


9.3286 


ordinate p - 


- 1.7 km 


log (A - B) 


9.7782n 


.*. from curve 


tp = 98.4 


( = ) e cos CO 7.8o88n 

\A-\-B ) 


7^ = J. 


D. 2416457.75 


log tan CO 


i.5i98n 


log const. 


4.1383 


CO 


91.73° 


logic 


1.6684 


sin CO 


9.9998 


logP 


2.0669 




/. 


k^ Vi - e" 


9.9902 


loge 


9.3288 


log a sin i 


7.8638 


e 


0.21 


a sin i 


73,000,000 + kni 



The preliminary values thus obtained are next tested by 
comparing the velocities derived from them by equations (10) 
with the observed velocities. To illustrate, let us compute the 
velocity for J. D. 24164^6.0, twenty days after the origin 
adopted in our curve. We have 



/ = 


24 


16496.0 


log cos {v -f co) 


9.8277n 


/- r = 




+ 38.0 




logic 


1.6684 


it-T) 




1.57978 






1. 496 in 


log/x 




0.48942 


A+B . . , 
cos(t' + co) 

2 


- 31.3 km 


M 




117.27° 


A-B 
2 




-0.3 


V 




136.01 


V 




+ 20.7 


f + CO 




227.74 


d^ 
dt 


= 


— 10.9 km 



» r is here taken one revolution earlier than the date for the periastron point marked 
on the curve. Using equation (8a), or (8b) we obtain T = J. D. 2416458.0 which is 
adopted. 



148 THE BINARY STARS 

In this manner we compute as many velocities as necessary to 
obtain a curve for comparison with the observed velocity curve. 
In the present instance this was done for every tenth day, and 
the results plotted as heavy black dots in the figure. By noting 
the discrepancies, it appears that the branch on the apastron 
side of the computed curve, if drawn, would be a little too 
sloping, the other branch too steep, which indicates that the 
computed value of e is a little too large. Changing this ele- 
ment and making the corresponding slight changes required 
in T and co, the test was repeated, and after a few trials, the 
following elements were adopted as best representing the ob- 
servations : 

V = -\- 21.9km 

P = 116.65 days 

e = 0.19 

K = 46.5 

0) = 96.23° 

r = J. D. 2416459.0 

a sin i = 73,000,000km 

The correction to the value of V was found last of all from 

the residuals of the final ephemeris by the simple formula 

[v] . 

^^, where n is the number of observations and v the residual, 

n 

o — c. The residuals from the final ephemeris and the final 
curve may be found in Lick Observatory Bulletin, No. 122, 1907. 
Lehmann-Filhes's method may be termed the classical one, 
and it is probably more generally used than any other. The 
method proposed by Rambaut is considerably longer and 
more involved than the later ones, and for that reason will not 
be described here. Wilsing's method, as originally published, 
was suitable only for orbits of small eccentricity, but Russell 
later extended it to make it applicable to larger eccentricities 
as well. This method is purely analytical, consisting in finding 
a Fourier's series for the velocity in terms of the elements. It 
should be very useful in special cases, particularly when the 
period is so nearly a year that one part of the velocity curve 
is not represented by any observations; but it is considerably 



THE BINARY STARS 149 

longer, in time consumed, than the method of Lehmann-Filh^s 
and other geometrical methods to be described presently, and 
it will not be further considered here. 

Certain features of the methods proposed by Schwarzschild 
and Zurhellen are both ingenious and practical. The following 
account of them is taken in substance from Curtis's article 
already referred to. 

schwarzschild' S METHOD 

Given the velocity curve and the period, Schwarzschild first 
determines the time of periastron passage. Let Mi and M2 be 
the observed velocities {i.e.j the velocities measured from the 
zero-axis) of maximum and minimum, and draw the line whose 

ordinate is — ^- ^. This line is the mean axis. Mark upon 

2 

it the points corresponding to P 1 2 and 3P J2 ; then lay a piece 
of semi-transparent paper over the plot, copy upon it the curve 
together with the mean axis and mark also the points o, P I2, 
P, and 3P/2. Shift the copy bodily along the mean axis for 
the distance P/2, and then rotate it 180° about this axis, — i.e., 
turn the copy face downward on the original curve keeping 
the mean axis in coincidence and bring the point o or P of the 
copy over the point P/2 of the original. The curves will then 
cut each other in at least four points, and, in general, in four 
points only. These will fall into two pairs, the points of each 
pair separated by an abscissa interval P/2. The points of one 
pair will be on different branches of the velocity curve, and it 
is easy to see that, if Vi and V2 represent their true anomalies, 
we shall have V2 = Vi -\- 180°. Now the only two points in the 
true orbit which are separated by one-half a revolution and for 
which at the same time this relation of the true anomalies 
holds are the points of periastron and apastron passage. 
Hence, to select these points, choose the two points of inter- 
section of the curve and its copy which are separated by half 
a revolution and which lie on different branches of the curve. 
To distinguish periastron from apastron we have the criteria: 
First, at periastron the velocity curve is steeper with respect 
to the axis than at apastron ; Second, the curve is for a shorter 



150 THE BINARY STARS 

time on that side of the mean axis on which the point of peri- 
astron lies. 

This method is exceedingly good except when the eccen- 
tricity is small. In this case co and T are quite indeterminate 
and small errors in drawing the velocity-curve will be very 
troublesome. The method of Lehmann-Filhes is then to be 
preferred. 

Having the value of T, the value of co is next found as fol- 
lows: From equations (la) and (3) it is readily seen that the 
position of the mean axis is 

= V -\- K e COS w = Fi , 

2 

and that, accordingly, the ordinate 2' of any point measured 
from the mean axis is 

z' = ^- Vi = Kcos(v-{- oj). (II) 

dt 

Now at periastron v = 0°, at apastron v = 180°. Hence, if 
we call the ordinates from the mean axis for these, points z'p 
and z'o we shall have 

z'p ^p — z' a 

COS CO = ^ or cos w = -^ (12) 

K 2K 

from which to determine co. This method is at its best when 
CO is near 90°. 

Zurhellen has simplified Schwarzschild's method of finding 
e, and we shall give this simpler form in connection with Zur- 
hellen's simple method of finding co. 

zurhellen's methods 

Zurhellen's method of determining co depends upon the rela- 
tions between the velocities for the two orbit points whose 
true anomalies are i 90°. From equation (11) we have, when 
V = - 90°, 

2i = + iC sin CO 

and when v = -\- 90°, 
Zi= — K sin CO. 



THE BINARY STARS 151 

Moreover, for these two points we have 
£1 = — £2 
Ml = - ikf 2 
(t,-T)= -{t2-T)', 

hence the two points are symmetrically placed with respect 
to the mean axis in the F-coordinate and with respect to the 
point of periastron passage in the -AT-coordinate. They may 
therefore be determined by rotating the curve copy through 
180° about the intersection of the ordinate of periastron with 
the mean axis, and noting the two points of intersection of the 
copy with the original curve. If the curve is prolonged through 
one and one-half revolutions, another point 180° from one of 
these, say at + 270°, can be determined in similar manner 
and the location of all three can then be checked by drawing 
the lines connecting the point z; = + 270° with v = — 90°, 
and V = — 90° with v = -\- 90°. These lines should cut the 
mean axis at its intersections with the ordinates of periastron 
and apastron respectively. From the ordinates of the two 
points V = =±=90°, measured from the mean axis, we have 

si — 22 ^ 21—22 , . 
sin CO = , or tan w = (12) 

2K Zp — Za 

from which to find w. The method is at its best when co is small. 

Zurhellen's simplification of Schwarzschild's method of find- 
ing e is also based upon the relations between the two points 
V = ^ 90°. Since 

tan 3^ £ = tan >^ i; tan (45° - H <t>) 
where (f> is the eccentric angle, we have, when v = ±90^ 

£1 = - (90° - 0), £2 = + (90° - ct>). 

Similarly, 

IT /'^^o , •, , sin sin (90° - 0) 

Ml = — (90° -- 0) + - 



»o 



o ^ ^ sin (^ sin (90° — </>) 



sin I 

sin (h s 

M2 = + (90° -<i>)- 

sin I 
and therefore 



M. - M. = 3^ 0. - <.) = (:8o° -2.t>)- "" ^"°° V^^ . (.3) 



Sin I 



152 



THE BINARY STARS 



The value of {h — /i) may be read off directly from the dia- 
gram, and the value of (90° — 4>) can then be taken from the 
table for equation (13), computed by Schwarzschild, which is 
given below. Like the above method for finding co this method 
is best when co is small. 



Schwarzschild' s Table for the Equation 
360' 



2 7; — sin 2 r; = 



{t2 - ti). 





t2 — tx 




t2 — ti 




t2 — tl 


V 


P 


V 


P 


V 


P 


0° 


0.0000 


30° 


. 0290 


60° 


0.1956 


I 


0.0000 


31 


0.0318 


61 


. 2040 


2 


. 0000 


32 


. 0348 


62 


0.2125 


3 


. 0000 


33 


. 0380 


63 


0.2213 


4 


O.OOOI 


34 


0.0414 


64 


0.2303 


5 


O.OOOI 


35 


. 0450 


65 


0.2393 


6 


. 0002 


36 


. 0488 


66 


0.2485 


7 


0.0004 


37 


0.0527 


67 


0.2578 


8 


. 0006 


38 


. 0568 


68 


0.2673 


9 


0.0008 


39 


0.061 1 


69 


0.2769 


10 


O.OOII 


40 


. 0656 


70 


0.2867 


II 


0.0015 


41 


0.0703 


71 


0.2966 


12 


. 0020 


42 


0.0751 


72 


0.3065 


13 


0.0025 


43 


. 0802 


73 


0.3166 


14 


0.0031 


44 


0.0855 


74 


0.3268 


15 


. 0038 


45 


0.0910 


75 


0.3371 


16 


0.0046 


46 


. 0967 


76 


0.3475 


17 


. 0055 


47 


0.1025 


77 


0.3581 


18 


. 0065 


48 


0.1085 


78 


0.3687 


19 


0.0077 


49 


O.II47 


79 


0.3793 


20 


0.0089 


50 


O.I2I2 


80 


0.3900 


21 


0.0103 


51 


0.1278 


81 


. 4008 


22 


0.0II7 


52 


0.1346 


82 


0.4II7 


23 


0.0133 


53 


O.I416 


83 


0.4226 


24 


0.0I5I 


54 


0.1488 


84 


0.4335 


25 


0.0170 


55 


O.I56I 


85 


0.4446 


26 


0.0I9I 


56 


0.1636 


86 


0.4557 


27 


0.0213 


57 


O.I713 


87 


. 4667 


28 


0.0237 


58 


0.1792 


88 


0.4778 


29 


. 0262 


59 


0.1873 


89 


0.4889 


30 


. 0290 


60 


0.1956 


90 


. 5000 



THE BINARY STARS 153 

Zurhellen also gives a method for finding the eccentricity 
by drawing the tangents to the curve at the points of perias- 
tron. These can be drawn quite accurately except when the 
periastron falls near a maximum or a minimum of the curve. 
Slight changes in its position will then introduce considerable 
changes in the inclinations of the tangent lines: 

The expression for the slope of a tangent may be written 

dx 2 IT dx _ 2 IT I dx 

~dt ~ ~F ' dM~ P ' i-ecos£* dE 

where ^ ( = ~r ) represents the ordinate drawn to the F-axis. 

Also, by introducing the known values 

cos E — sin <}) . cos <^ sin £ 

cos V = , sin z; = 

I — e cos E I —e cos E 

and transforming and simplifying we may write the funda- 
mental equation (3) in the form 

dz __ , cos <f> cos CO cos £ — sin oj sin E 

X = — = K cos <p . • 

dt I — e cos E 

Differentiating with respect to E, substituting and reducing, 
we have 

dx 2 TT ^, . — cos (f> cos CO sin £ — sin co cos £ + ^ sin co , . 

— =—Kcos<t>. — . (14) 

dt P (i — ecosEY 

At periastron E = 0° and at apastron E = 180°, whence we 
have 



dx — 2 TT K COS sin CO dx _ -\- 2 w K cos (f) sin co 



dtp P{i - eY ' dta P (i + eY 



and therefore 

dx I dx (i + eY 



dtp I dta (i - eY 

whence 

9 + » 



«'. 



154 THE BINARY STARS 

king's method 

The methods of orbit computation so far described in this 
chapter all rest upon the curve drawn to represent as closely 
as possible the observed velocities and, at the same time, to 
satisfy the conditions for elliptic motion. Unless the measures 
are very precise, the first approximation will ordinarily not be 
satisfactory. As stated on page 140, the only remedy is to 
compute an ephemeris from the elements and, on the basis of 
the residuals thus found, to draw a new curve. This process 
is sometimes repeated three or four times before a curve is 
found which will yield elements upon which a least squares 
solution may be based. 

The method devised by Dr. King, which is now to be pre- 
siented, aims to substitute a rapid graphical process for testing 
the preliminary curve. Dr. King shows that a circle having 
its center on the mean axis and a radius equal to K, the semi- 
amplitude of the velocity curve, "may be used as the equiv- 
alent of the hodograph of observed velocities." ^° 

Let the velocity curve and the circle be drawn (see Figure 9) 
and the abscissa distance corresponding to one revolution (P 
being assumed to be known) be divided into any convenient 
number of parts, say forty. ^^ Now mark consecutive points 
on the circumference of the circle by drawing lines parallel to 
the mean axis at the intersections of the velocity curve with 
the ordinates corresponding to successive values of the ab- 
scissa and extending them to the circle. The circumference 
will be divided into forty unequal parts, but these inequalities 
will be found to vary uniformly. "The points will be close 
together in the vicinity of one point of the circle, and will 
gradually separate as we proceed in either direction therefrom, 
until at the diametrically opposite point they reach their maxi- 
mum distance apart." These unequal arcs of the circle cor- 
respond to the increase in the true anomalies in the orbit in 
the equal time intervals, and therefore the point of widest 

10 For the proof of this relation the reader is referred to the original article, Astrophysical 
Journal, vol. 27, p. 125, 1908. 

11 An even number should be chosen, and it is obviously most convenient to make the 
drawing upon coordinate paper. 



THE BINARY STARS 155 



s 



I 



^ 



s 



^ 



^ 



% 



i 



"i 



\ 



% 



h 



n. 



?A 



^ 



n 



I 



\ 



^ 



X 



t 



h 



\ 




s 



\ 



:s 



21 



iS? 



Fpr- 



Figure 9. King's Orbit Method. Graph for « = 0.75, w = 60° 



156 THE BINARY STARS 

separation of the circle divisions corresponds to periastron, 
that of least separation, to apastron. Further, the angle be- 
tween the F-axis and the periastron-half of the diameter 
between these two points is equal to co. To locate the point of 
periastron on the velocity curve, find the intersection of the 
steeper branch of the curve with a line drawn from the peri- 
astron point on the circle parallel to the mean axis. 

It is evident that the division points of the circumference 
will be symmetrically disposed with respect to the apsidal 
diameter (the diameter joining periastron and apastron points) 
only when one of the division points in the line of abscissae 
corresponds to an apse. In general, the periastron point will 
lie within the longest division of the circumference, the apas- 
tron point within the shortest. If desired, the approximate 
position of one of these points may be used as a new origin 
from which to set off the fortieths of the period along the axis 
of abscissae, and two division points on the circle may then be 
brought into closer coincidence with the apsidal points. 

Since — - varies inversely as the square of the distance from 
at 

the focus, by measuring the lengths di and d2 of the arcs at 

points where v equals Vi and V2, we have 

di _ {i -\- e cos viY 
di {i -\- e cos ^2)^ 

and hence if the arcs are measured at the points of periastron 
and apastron where v respectively equals 0° and 180°, 



d2 \i - ej 



V di— V d2 

V di-^ V~d, 



or e = = • (16) 



which determines e. 

It is generally sufficiently accurate to measure the chords 
instead of the arcs; when the eccentricity is high and the arcs 
at periastron inconveniently long, additional points of division 
may readily be inserted. 

It will be observed that this process furnishes a more thorough test 
of the accuracy of the graph (velocity curve) than the method of 
equality of areas. If it is imperfect, the points on the circumference 



THE BINARY STARS I57 

of the circle will not be distributed according to the regular order of 
increase or decrease of the included arcs. If an ordinate of the graph 
is too long or too short, the corresponding point on the circumfer- 
ence will be too near to or too far from the vertical diameter. If the 
points of maximum and minimum velocity have not been well deter- 
mined, the diameter of the circle will be too long or too short. In the 
former case all the points of the circumference will be crowded away 
from the vertical diameter; in the latter, toward it.^^ 

To test a given set of elements by comparison with the ob- 
servations proceed as follows : 

Construct a circular protractor on some semi-transparent 
material (e.g., celluloid or linen tracing cloth) and divide it 
into forty parts by radii to points on the circumference repre- 
senting the true anomalies for the given value of e correspond- 
ing to every 9° of mean anomaly {i.e., to fortieths of the period). 
If the eccentricity is less than 0.77 the values of the true anom- 
aly can be taken directly from the Allegheny Tables. 

On the plot of the orbit draw a circle of radius K with its 
center on the mean axis and draw its vertical diameter. Set 
the protractor upon the circle, making the centers coincide, 
and turn the apsidal diameter of the protractor until it makes 
an angle equal to co with the vertical diameter. Now note the 
points where the radial lines representing the anomalies inter- 
sect the circumference of the circle. The abscissa axis of the 
plot also having been divided into forty equal parts, erect 
perpendiculars at the points of division equal to the corres- 
ponding ordinates of the circle. A freehand curve through the 
extremities of these perpendiculars (i.e., ordinates to the mean 
axis) gives the computed curve or 'ephemeris', and the resid- 
uals can be read directly from the plot. The advantage of 
using coordinate paper will be obvious. 

From the account just given it will appear that King's 
method is longer, or at least not shorter, than the others de- 
scribed if only a single orbit is to be computed. But when 
orbit computation is to be taken up as a part of a regular pro- 
gram of work, the method has very decided advantages. It 
is then to be used as follows: 

12 King, loc. cit. 



158 THE BINARY STARS 

Let a set of protractors be constructed on transparent cellu- 
loid with radii representing the divisions in true anomaly for 
every 9° of mean anomaly for the values e = 0.00, e = 0.05 
to g = 0.95. 

With the aid of these protractors draw curves on tracing 
linen representing orbits with all values of e from 0.00 to 0.95 
and all values of co from 0° to 360°. The intervals for e should 
be 0.05, save for the larger values which are seldom used, and 
for CO, 15°. Practically, values of w to 90° will suflfice, the curves 
for the values in the remaining quadrants being obtained by 
inverting the sheet, and by looking through the linen from the 
back in the two positions. Given the protractors, a complete 
set of curves may be constructed in about ten hours' time. 

Having such a set of curves, plot the normal places for any 
given binary star on the same scale as these curves in time and 
in velocityP Now place the standard curves upon the plot 
until one is found that fits the observations. "If two or more 
curves seem to give about equally good representations, it is 
quite possible to interpolate elements between the graphs 
plotted." 1* 

By this process values of e correct to within one or two 
hundredths and of co correct within a few degrees can generally 
be obtained at the first trial and with an expenditure of less 
than ten minutes' time. The time of periastron passage fol- 
lows at once, and this set of preliminary elements may then 
be used as the basis for a least squares adjustment. The pro- 
cedure has been found very satisfactory at the Dominion 
Observatory at Ottawa. 

Russell's short method 

Professor Henry Norris Russell has devised a graphical 
method which is equally simple in its practical application.^^ 

" Since the velocity curve is ordinarily based on from fifteen to twenty normal places 
the work of multiplying by the appropriate reduction factors will require a very few minutes 
only. Of course the amplitude of the curve as well as the period must be known before 
the reduction factors can be obtained. These are known with sufficient accuracy from the 
preliminary plots. 

" R. K. Young. Orbit of the Spectroscopic Binary 2 Sagittae. Journal R. A. S.C., vol. 
17, p. 131. 1917. 

^* Aslrophysical Journal, vol. 40, p. 282, 1914. 



THE BINARY STARS I59 

Write equation (la) in the form 

dz 
P = F + -— = F + X e cos CO + ^ cos (v + w) =G + i^ cos (v + co) 

^' (17) 

where p represents the observed radial velocity. 

Then {G + K) is the maximum, (G — K) the minimum 
value of the velocity so that G and K may be estimated at 
once from the freehand curve. The period is also assumed 
to be known. Equation (17) may then be written in the form 

P-G 



cos {v + co) 



(18) 



and the value of (v + co) computed for each observed value of p. 

If we subtract the corresponding values oi M -\- Mo from each of 
these, we shall have values of {v — M) -\- {oj — Mo). The second part 
of this expression is constant, while the first is the equation of the 
center in the elliptic motion. During a revolution this varies between 
equal positive and negative limits which depend only on the eccen- 
tricity, and are nearly proportional to it, as is shown in the following 
table. 



Eccentricity 


O.IO 


0.20 


0.30 


0.40 


0.50 


0.60 


0.70 


0.80 


0.90 


Maximum 




















equation 




















of center 


ii°.5 


23°.o 


34°.8 


46°.8 


S9°.2 


72^3 


86°.4 


loa^.s 


I22°.2 



If the values of v — M -{- a — Mo are plotted against those of 
M -i- Mo, we obtain a diagram which, since it represents the relations 
between the mean and the true anomalies, we may call the anomaly 
diagram. If on this diagram a curve is drawn to represent the plotted 
points, half the difference between its maximum and minimum ordi- 
nates will be the greatest value of the equation of the center, from which 
e may be found at once by means of the table. The mean of the maxi- 
mum and minimum ordinates will be the value of w — Mo. The 
instants when v — M -\- oi — Mo has this value are those of periastron 
and apastron passage, the former corresponding to the ascending branch 
of the curve, which is always the steeper. The abscissae of the corres- 
ponding points of the curve are Mo and Mo + 180°. The values of e. 
Mo, and co are now known, and the remaining elements may be found 
at once from K and G. 



l60 THE BINARY STARS 

According to Russell the "principal advantage of this method 
is that the form of the curves which give v — M sls Si function 
of M depends upon e alone." For further details and an illus- 
trative example we refer the reader to the original memoir. Up 
to the present time the method has not come into general use. 

A similar remark applies to the graphic method proposed 
by Dr. K. Laves, in 1907, and, of course, to the two short 
methods quite recently proposed by Dr. F. Henroteau, which 
take advantage of the Allegheny Tables of Anomalies in a 
novel way. References to the original papers containing these 
methods are given at the end of this chapter. 

DIFFERENTIAL CORRECTIONS TO THE ELEMENTS 

Whatever method may be used in finding the preliminary 
orbit it is generally advisable to determine the correction to the 
elements by the method of least squares.^^ The formula derived 
by Lehmann-Filhes from which the coefficients for the observa- 
tion equations are to be computed may be written as follows: 

d — = d V -{■ [cos {v -\- 03) -\- e cos co]d K 
dt 

sin(i' + co)sinr 

■f- K [cos CO {2 -\- e cos v)\ a e 

I — e^ 

— K [sin {v -^ 0)) -\- e sin 03] dc^ 

-\- sin {v -\- co) {i -\- e cos vY ,1 ^ I' 

(i - e2) '' 

- sin (t; -f o)) (I -f e cos v^ ,,, {i - T)d ix . (19) 

(i — e) '^ 

"^^ Publication Allegheny Observatory, vol. i, p. 33, 1908. 

The advantages of applying the method of least squares to the definitive solution of 
spectroscopic binary stars have been clearly stated by Schlesinger in this paper. "The 
problem," he says, "involves the evaluation of five quantities (six if the period be included) 
that are so interwoven as to make their separate determination a matter of some difficulty. 
Herein lies the chief advantage for this case of the method of least squares; for it enables 
us to vary all of the unknowns simultaneously instead of one or two at a time. . 
Further, the method of least squares not only has the advantage of greater accuracy and 
of telling us how reliable our results are, but it eliminates from the computations any 
personal bias or arbitrary step . . . the method should be used in almost every case 
where the elements are not avowedly provisional." Not all computers are so enthusiastic 
as to the method. Judgment must of course be exercised in all orbit work as to whether 
the data at hand warrant anything beyond the computation of purely provisional elements. 
In spectroscopic binary orbits, for example, such factors, among others, as the number and 
quality of the plates and their distribution over the velocity curve, the character of the 
star's spectrum, and the character of the orbit must be considered in making this decision. 



THE BINARY STARS l6l 

In practice the period is almost always assumed to be known 
with accuracy and the last term of the equation is omitted. 

To facilitate the computation Schlesinger has transformed 
this equation as follows: 

Put 

a = 0.452 sin V {2 -\- e cos v) 

{i -\- e cos vY 
p = 

(I + cY 

T = d V -\- e cos ixi d K -\- K cos w d e — K e swujj d 03 

K = dK 

TT = — K do) 

zr 2-21 , 

€ = — K de 

I - e^ 



\ I - c 



T = /fyL, J-LX£. J!-iT 



e I — e 



ni= - K J-tl . ^L dfi, and u = {v + co). 
\ I — e i — e 

Then the equations of condition take the form 

d^ 
d — = r + cos u ' K -\- sinu- tt + a sin « • € 

^^ + ^ sinu- T-\- ^ sin u- {t- T)m. (20) 

The quantities a and /8 can be tabulated once for all and such 
a tabulation is given by Schlesinger {loc. cit,) so arranged "as 
to render the normal equations homogeneous and to enable all 
multiplications to be made with Crelle's tables without inter- 
polation." If this notation is used, the computer should have 
these tables at hand.^^ 

When both spectra are visible on the plates, the orbits for 
the two components with respect to the center of mass may be 
determined separately. It is obvious that the two sets of 
values of F, e, T, and P must be identical, the values of w 
must differ by 180°, while the two values for K depend upon 

" Dr. R. H. Curtiss has shown that this formula and therefore the least squares solu- 
tion can be made appreciably shorter. (Publications Astronomical Observatory, University 
of Michigan, Vol. II, p. 178, 1916). 



l62 THE BINARY STARS 

the relative masses of the components. The preliminary ele- 
ments for the two components, when independently deter- 
mined, will, in general, not harmonize perfectly. To obtain 
the definitive values the best procedure is the one first sug- 
gested, I believe, by Dr. King.^^ It consists in combining all 
the observations, those for the secondary with those for the 
primary, into a single set of observation equations (equations 
of condition) and solving for one complete set of elements. If 
we write co' = (co + i8o°) and distinguish the values of K for 
the two components by writing Ki and K2 respectively, the 
equations in the notation of (19) assume the form 

d — = dV -\- [cos(r + co) -f- e cosco]Jivi+ [cos(f + co') + ecosco'j dKi 
di 



n sin (v + 03) sin v J j^ 

-h \ cosco {2 -\- e cos v) \ K\ 

LI i-^' J 

f , sin(t;+a)')sinT; n 1 1^ 1 j 

+ \ cos CO (2 + e cos ij) \ KiXde 

— I I sin (r + co) + e sin CO \ K\-]r \ sin (r + co') + ^ sin co' \ KAdw 

+ sin (t» + co) (i 4- ^ cos vY Ki 

1 M 
+ sin {v + co') (I + e cos v)^ K^ —-^^ ,^^ d T, (21) 

the value of the period being assumed to require no correction. 
Since K2 does not affect the residuals of the primary com- 
ponent, nor Ki those of the secondary, the terms dK^ and dKi 
disappear from the equations representing the residuals from 
the primary and secondary curves, respectively. 



SECONDARY OSCILLATIONS 

When the orbit of a spectroscopic binary star has been com- 
puted and the theoretical velocity curve drawn, it is some- 
times found that the observed normal places are so distributed 

i«See Harper's paper, in Publications of the Dominion Observatory, Vol. i, p. 327. I9i4- 
Dr. Paddock independently developed an equivalent equation. Lick Observatory Bulletin, 
Vol. 8, pp. 156, IS7. 1915- 



THE BINARY STARS 



163 



with respect to the curve representing simple elliptic motion 
as to suggest that a secondary oscillation is superimposed upon 
it. The question is whether this grouping arises from some 
source of error in the measurement of the spectrograms, from 
erroneous values of one or more of the orbit elements, or from 
a real oscillation such as might be produced, for example, by 
the presence of a third body in the system. This question has 
been discussed by Schlesinger and Zurhellen, and later by 
Paddock. Schlesinger and his associates at the Allegheny 
Observatory have shown that 'the blend eflfect' caused by the 
overlapping of the absorption lines of the two component 



:_^^i 


-■i'^% 


*i^ -(ST '■ 


-it s 


"E ^^-^.^^ 


' : ^^5^^^ 



Figure 10. Radial Velocity Curve of f Geminorum. The secondary oscillation 

is probably real 



Spectra "may produce such an apparent oscillation." They 
have also shown that it may be produced by chance errors in 
the velocities derived from the different lines of the spectrum, 
and they are convinced that a critical analysis of the data will 
dispose of a considerable percentage of cases wherein secondary 
oscillations have been suspected. The possibility of a real 
secondary oscillation must of course be recognized, and when 
a full analysis has shown that such an oscillation is present, 
additional terms may be introduced into the equations of con- 
dition to represent it upon the assumption that it is produced 
by a third body revolving in a circular orbit about one of the 
other two components. In the cases that have arisen thus 
far this simple assumption has yielded a satisfactory represen- 
tation of the data, though it is apparent that there is no 
reason for limiting such additional bodies to circular orbits. 
Let T' represent the time when the secondary curve crosses 
the primary from below, K' the semi-amplitude of the sec- 
ondary oscillation, m' the ratio of the principal period to that 



l64 THE BINARY STARS 

of the secondary oscillation, assumed to be known (it is gen- 
erally taken to be an integer), and put u' = m'li {t — T'), 
T = — m' jjlK' dT',K' = dK^; then the additional terms re- 
quired in equation (20) are 

+ sin u' • k' -\- cos u' ' t' . (22) 

For a more complete discussion of secondary oscillations the 
reader is referred to the articles cited above. 

ORBITS WITH SMALL ECCENTRICITY 

In a circular orbit the elements T and w obviously have no 
significance, and when the eccentricity is very small they 
become practically indeterminate by the geometrical or graph- 
ical methods which have here been described. Further, if 
approximate values are assumed, it is impossible to find cor- 
rections to both elements from the same least squares solution 
because the coefficients for the differential corrections will be 
nearly or quite equal. Some computers have overcome this 
difficulty by assuming the preliminary value of cu as final, and 
determining corrections to 7", but this is hardly a solution of 
the problem. In such orbits the analytic method possesses 
great advantages, as has been shown by several investigators, 
notably Wilsing and Russell, Zurhellen, and Plummer. Pad- 
dock has quite recently examined the question in great detail, 
extending some of the earlier developments and adapting them 
for computation. A full account of these methods would re- 
quire more space than is available here, and it has seemed best 
to refer the reader to the original papers. 

REFERENCES 

The following list of papers relating to one phase or another 
of the computation of orbits of spectroscopic binary stars, while 
not exhaustive, contains most of the important ones. 

Rambaut, a. a. On the Determination of Double Star Orbits from 
Spectroscopic Observations of the Velocity in the Line of Sight. 
Monthly Notices, Royal Astronomical Society, vol. S^^ P- 3 16, 1891. 



THE BINARY STARS 165 

WiLSiNG, J. tJber die Bestimmung von Bahnelementen enger Doppel- 
sterne aus spectroskopischen Messungen der Geschwindigkeits- 
Componenten. Astronomische Nachrichten, vol. 134, p. 90, 1893. 

Lehmann-Filhes, R. Cber die Bestimmung einer Doppelsternbahn 
aus spectroskopischen Messungen der in Visionradius liegenden 
Geschwindigkeits-Componente. Astronomische Nachrichten, vol. 
136, p. 17, 1894. 

ScHWARZSCHiLD, K. Ein Verfahren der Bahnbestimmung bei spec- 
troskopischen Doppelsternen. Astronomische Nachrichten, vol. 752, 
p. 65, 1900. 

Russell, H. N. An Improved Method of Calculating the Orbit of a 
Spectroscopic Binary. Astrophysical Journal, vol. i^, p. 252, 1902. 

. A Short Method for Determining the Orbit of a Spectroscopic 

Binary. Astrophysical Journal, vol. 40, p.. 282, 19 14. 

NijLAND, A. N. Zur Bahnbestimmung von spektroskopischen Dop- 
pelsternen. Astronomische Nachrichten, vol. 161, p. 105, 1903. 

Laves, K. A Graphic Determination of the Elements of the Orbits of 
Spectroscopic Binaries. Astrophysical Journal, vol. 26, p. 164, 1907. 

Zurhellen, W. Der spectroskopische Doppelstern Leonis. Astro- 
nomische Nachrichten, vol. 1^3, p. 353, 1907. 

. Bemerkungen zur Bahnbestimmung spektroskopischer Dop- 



pelsterne. Astronomische Nachrichten, vol. 775, p. 245, 1907. 
. Weitere Bemerkungen zur Bahnbestimmung spektroskopi- 



scher Doppelsterne — u. s. w. Astronomische Nachrichten, vol. 177, 
p. 321, 1908. 

. Uber sekondare Wellen in den Geschwindigkeits-Kurven 



spectroskopischer Doppelsterne. Astronomische Nachrichten, vol. 
187, p. 433, 191 1. 

King, W. F. Determination of the Orbits of Spectroscopic Binaries. 
Astrophysical Journal, vol. 27, p. 125, 1908. 

Curtis, H. D. Methods of Determining the Orbits of Spectroscopic 
Binaries. Publications A.S.P., vol. 20, p. 133, 1908. (This paper 
has, with the author's permission, been very freely used in preparing 
my chapter on the subject.) 

Plummer, H. C. Notes on the Determination of the Orbits of Spectro- 
scopic Binaries. Astrophysical Journal, vol. 28, p. 212, 1908. 



l66 THE BINARY STARS 

ScHLESlNGER, F. The Determination of the Orbit of a Spectroscopic 
Binary by the Method of Least Squares. Publications of the Alle- 
gheny Observatory, vol. i, p. 33, 1908. 

. On the Presence of a Secondary Oscillation in the Orbit of 

30 H Ursae Majoris. Publications of the Allegheny Observatory, vol. 2, 
p. 139, 191 1. 

. A Criterion for Spectroscopic Binaries, etc. Astrophysical 



Journal, vol. 41, p. 162, 1915. 

Paddock, G. F. Spectroscopic Orbit Formulae for Single and Double 

Spectra and Small Eccentricity. Lick Observatory Bulletin, vol. 8, 

p. 153, 1915. 
CuRTiss, R. H. Method of Determining Elements of Spectroscopic 

Binaries. Publications of the Astronomical Observatory, University 

of Michigan, vol. 2, p. 178, 1916. 

Henroteau, F. Two Short Methods for Computing the Orbit of a 
Spectroscopic Binary Star by Using the Allegheny Tables of 
Anomalies. Publications A. S. P., vol. 2Q, p. 195, 191 7. 



CHAPTER VII 

ECLIPSING BINARY STARS 

We have seen that one of the first binary systems to be dis- 
covered with the spectrograph was Algol (jS Persei), long known 
as a variable star. There are other stars whose light varies in 
the same peculiar manner as that of Algol; that is, while it 
remains sensibly constant at full brightness the greater part 
of the time, at regular intervals it fades more or less rapidly 
to a certain minimum. It may remain constant at this mini- 
mum for a short time and then recover full brightness, or the 
change may be continuous. In either case the entire cycle of 
change is completed in a small fraction of the time of constant 
light between the successive minima. 

The hypothesis that in every such case the star, as viewed 
from the Earth, undergoes a total, annular, or partial eclipse, 
the eclipsing body being a relatively dark star revolving with 
the other about a common center of gravity, completely ac- 
counts for the observed facts and has been proved to be 
correct not only in the one instance, Algol, but also in that of 
every Algol-type variable which has been investigated with 
the spectrograph. Undoubtedly it is the correct explanation 
for all stars of this type; they are all binary systems. 

Unless the darker star is absolutely non-luminous, there 
should be a second minimum when the bright star passes 
between it and the Earth, the relative depth of the two minima 
depending upon the relative intensity of the light of the two 
stars and upon their relative areas. Such a secondary mini- 
mum has been observed in jS Lyrae and in this star the light 
is not quite constant at any phase, either maximum or min- 
imum. There is now no doubt but that this star and others 
like it are also binary systems. 

It was formerly thought that a distinction could be drawn 
between variable stars of the type of Algol and those of the 



l68 THE BINARY STARS 

type of /3 Lyrae, but measures with sensitive modern photo- 
meters, such as the selenium-cell, the photo-electric cell, and 
the sliding-prism polarizing photometers, and measures of 
extra-focal images on photographic plates have attained such 
a degree of accuracy that a variation considerably less than 
one-tenth of a magnitude can be detected with certainty; and 
it now appears that Algol itself not only has a slight secondary 
minimum but that its light is not quite constant at maximum. 
The distinction, therefore, breaks down and we may regard 
all the stars of these two types as members of a single class, 
calling them eclipsing binaries or eclipsing variables according 
to the point of view from which we take up their investigation. 
There are, in all, more than 150 eclipsing binary stars known 
at the present time and a large percentage of them are too 
faint to photograph with our present spectrographic equip- 
ment. It is therefore a matter of great interest to inquire what 
information, if any, as to the orbits of these systems can be 
derived from their light curves, the curves, that is, which are 
constructed by taking the observed stellar magnitudes as 
ordinates and the corresponding times as abscissae. 

Professor E. C. Pickering ^ made an investigation of the 
orbit of Algol on the basis of its light curve as early as 1880, 
and showed that a solution of such orbits was possible if certain 
reasonable assumptions — for example, that the two stars are 
spherical with uniformly illuminated disks and move in cir- 
cular orbits — were granted. The subject was resumed by him 
later, and was taken up also by Harting, Tisserand, A. W. 
Roberts, and others. But the most complete investigation so 
far is that begun at Princeton University, in 191 2, by Russell 
and Shapley, the theoretical part being contributed, mainly, 
by the former, the application to particular systems, mainly, 
by the latter. The present chapter will be based entirely upon 
this investigation. 

In the most generaJi, case the problem is an extremely com- 
plicated one, for the orbits must be regarded as elliptical with 
planes inclined at a greater or less angle to the line of sight, 

1 Dimensions of the Fixed Stars, with special reference to Binaries and variables of 
the Algol Type. Proceedings American Academy of Arts and Sciences, vol. i6, p. i, 1881. 



THE BINARY STARS 169 

and the two components as ellipsoids, the longest diameter of 
each being directed toward the other star. Moreover, the 
disks may or may not be uniformly illuminated ; they may be 
darker toward the limb, as our own Sun is, the degree of dark- 
ening depending upon the depth and the composition of the 
enveloping atmosphere, and the side of each which receives 
the radiation from the other may be brighter than the opposite 
side. The complete specification of an eclipsing binary system 
therefore requires a knowledge of at least thirteen quantities, 
which in Russell's notation, are as follows: 



Orbital Elements 




Eclipse Elements 


Semi-major axis 


a 


Radius of larger star r\ 


Eccentricity 


e 


Radius of smaller star r » 


Longitude of periastron 


Oi 


Light of larger star Li 


Inclination 


i 


Light of smaller star L2 


Period 


P 


and at least 3 constants defining 


Epoch of principal conjunction 


to 


the amount of elongation, of dark- 



ening at the limb, and of brighten- 
ing of one star by the radiation of 
the other. 

The longitude of the node must remain unknown, as there is no hope 
of telescopic separation of any eclipsing pair. 

The value of a in absolute units can be found only from spectroscopic 
data. In the absence of these, it is desirable to take a as an unknown 
but definite unit of length, and express all other linear dimensions in 
terms of it. Similarly, the absolute values of L\ and L2 can be deter- 
mined only if the parallax of the system is known. But in all cases the 
combined light of the pair, Li + L2, can be taken as the unit of light 
and the apparent brightness at any time expressed in terms of this. 
This leaves the problem with eleven unknown quantities to be deter- 
mined from the photometric measures. Of these, the period is invari- 
ably known with a degree of accuracy greatly surpassing that attainable 
for any of the other elements, and the epoch of principal minimum can 
be determined, almost independently of the other elements, by inspec- 
tion of the light-curve. Of the remaining elements, the constants 
expressing ellipticity and 'reflection' may be derived from the observed 
brightness between eclipses. These effects are often so small as to be 
detected only by the most refined observations. The question of 
darkening toward the limb may well be set aside until the problem is 
solved for the case of stars which appear as uniformly illuminated disks. 



170 THE BINARY STARS 

This leaves us with six unknowns. Fortunately, systems of such 
short period as the majority of eclipsing variables have usually nearly 
circular orbits (as is shown both by spectroscopic data and by the 
position of the secondary minimum). The assumption of a circular 
orbit is therefore usually a good approximation to the facts, and often 
requires no subsequent modification. 

Russell's papers discuss first the simplified problem: 

Two spherical stars, appearing as uniformly illuminated disks, and re- 
volving aboul their common center of gravity in circular orbits, mutually eclipse 
one another. It is required to find the relative dimensions and brightness of 
the two stars, and the inclination of the orbit, from the observed light-curve. 

Even in this form four different cases are presented, de- 
pending upon the character of the photometric data and the 
completeness of the observations; only one minimum or both 
minima may have been adequately observed, and these may 
or may not show a constant phase. To illustrate the method 
of investigation, I shall here present (by permission, almost 
entirely in Russell's own words) the two simplest cases, namely, 
those in which a constant phase has been observed in one or 
both minima and in which the eclipses are therefore either 
total or annular, and shall refer the reader who wishes to 
pursue the subject to the original memoirs. 

We may assume P and to as already known. If the radius of the rela- 
tive orbit is taken as the unit of length, and the combined light of the 
two stars as the unit of light, we have to determine four unknown 
quantities. Of the various possible sets of unknowns, we will select 
the following: 

Radius of the larger star n 

Ratio of radii of the two stars k 

Light of the larger star Li 

Inclination of the orbit i 

The radius of the smaller star is then r2 = kri, and its light, L2 = i — Li. 
It should be noticed that, with the above definitions, k can never exceed 
unity, but L2 will exceed Li whenever the smaller star is the brighter 
(which seems to be the fact in the majority of observed cases). 

We will suppose that we have at our disposal a well-determined 
'light-curve', or more accurately, magnitude-curve. . . . From 



THE BINARY STARS 171 

this we can pass at once to the intensity-curve, giving the actual light- 
intensity / as a function of the time, by means of the equation 

log/ = o.4(wo — w), (i) 

where Wo is the magnitude during the intervals of constant light be- 
tween eclipses (which is determined with relatively great weight by 
the observations during these periods and, like P, may be found once 
for all before beginning the real solution). This of course expresses / in 
terms of our chosen unit Li + Li. 

Such a magnitude-curve or intensity-curve will in general show two 
depressions, or 'minima', corresponding to the mutual eclipses of the 
two components. Under the assumed conditions, it is well known: 

1. That the two minima will be symmetrical about their middle points' 
and that these will be separated by exactly half the period. 

2. If the eclipse is total or annular, there will be a constant phase at 
minimum during which the magnitude- or intensity-curve is horizontal; 
but if the eclipse is only partial, this will not be the case. 

3. The two minima will be of equal duration, but usually of unequal 
depth. At any given phase during one minimum one of the stars will 
eclipse a certain area of the apparent disk of the other. Exactly half a 
period later, at the corresponding phase during the other minimum, the 
geometrical relations of the two projected disks will be the same, except 
that now the second star is in front, and eclipses an equal area — though 
not an equal proportion — of the disk of the first. The intensity-curves for 
the two minima must therefore differ from one another only as regards 
their vertical scales, which will be in the ratio of the surface intensities of 
the two stars. 

4. The deeper (primary) minimum corresponds to the eclipse of the star 
which has the greater surface intensity by the other. Whether this is the 
larger or smaller star must be determined by further investigation. 

Suppose that at any time during the eclipse of the smaller star by 
the larger the fraction a of its area is hidden. The light received from 
the system at this moment will be given by the equation 

/i = I - aLz. (2) 

Half a period later, an equal area of the surface of the larger disk, and 
hence the fraction k'^o. of its whole area will be eclipsed. The observed 
light will then be 

l2= \- k^aLi. (3) 

Since Li -f L2 = i, we find at once from these equations 

(I - k) + ^-?-^ = a. (4) 



172 THE BINARY STARS 

If both minima have been observed and show constant phases, the 
eclipse of the smaller star by the larger is total and the other eclipse 
annular; in both cases an area equal to the whole area of the smaller 
star is obscured; that is, a = i. If Xi and X2 are the values of the ob- 
served intensities during the constant phase at the two minima, we 
have by (4) 

*' = '-^- (5) 

Moreover, by (2), Xi = i — L2 = Li. The brightness of the two stars 
and the ratio of their radii, are thus determined, leaving only ri and i 
to be found. 

There are, however, two solutions with different values of k accord- 
ing as we regard the principal or secondary minimum as total. We 
shall see later how we may distinguish the correct solution in a given 
instance. 

[In case only the primary minimum has been observed], if the ob- 
served minimum intensity is X and we assume that the observed eclipse 
is total, we have from (2), L2 = I — X; if annular, (3) gives k'^Li = 
I — X. In either case, for any other value / of the observed intensity, 

a = ^' (6) 

I— X 

We thus know a as a function of the time, and from this have to deter- 
mine k, ri, and i. 

Take the center of the larger star as origin, and let 6 be the true 
longitude of the smaller star in its orbit, measured from inferior con- 
junction. Then ^ 

e=^{t-to). (8) 

From the light-curve and (6) we can find the value of a for any value 
of 0y or vice versa. Now a, which is the fraction of the area of the 
smaller disk which is eclipsed at any time, depends on the radii of the 
two disks, and the apparent distance of their centers, but only on the 
ratios of these quantities (being unaffected by increasing all three in 
the same proportion). If 5 is the apparent distance of centers, we have 
therefore 



-'{^■ry-i'-v) 



2 For convenience, I have preserved the equation numbers as given in Russell's original 
paper. His equation (7) is omitted because it does not relate to the cases here discussed. 



THE BINARY STARS 173 

where/ is a function, the details of calculation of which will be discussed 
later. 

For any given value of k we may invert this function, and write 

b 

= (^, a). (9) 

This function, or some equivalent one, may be tabulated once for all 
for suitable intervals of k and a. 

By the geometry of the system, we have 

52 = sin2 Q _j_ cos^ i cos2 e = cos^ i + sin^ i sin" d, (10) 

whence 

cos" i + sin" i sin" 6 = ri" {^(/^.a)}". (11) 

Now let ai, a^, as be any definite values of a and Bu 62, Oz the cor- 
responding values of 6 (which may be found from the light-curve). 
Subtracting the corresponding equations of the form (11) in pairs, and 
dividing one of the resulting equations by the other, we find 

sin" di - sin" 62 = [<i> (k, ai) }"- {0 (ife.aa)}^ 

f ^ f 7- = \f/(k, 01, a2, 03). (12) 

sin" 02 - sin" ds = {<t> {k, az) }"- {</. (k, aa) }' 

The first member of this equation contains only known quantities. 
The second, if ai, a2, and as are predetermined, is a function of k alone. 
If this function is tabulated, the value of k in any given case can be 
found by interpolation, or graphically. The equations (11) can then 
be used to find ri and i. 

A theoretical light-curve may then be found, which passes through 
any three desired points on each branch of the observed curve (assumed 
symmetrical). These points may be chosen at will by altering the 
values of ai, a2, and as. In practice it is convenient to keep az and as 
fixed, so that \J/ becomes a function of k and ai only, and may be tabu- 
lated for suitable intervals in these two arguments. This has been 
done in Table II, in which a2 is taken as 0.6 and as as 0.9. If ^ = sin^ $2, 
B = sin^ 02 — sin^ 63, (12) may be written 

sin" di = A-^ B^p(k, ai). (13) 

The points a and b on the light-curve corresponding to a2 and as, to- 
gether with the point corresponding to any one of the tabular values 
of ai, then give a determination of k. By taking a suitably weighted 
mean of these values of k, a theoretical light-curve can be obtained 
which passes through the points a and b, and as close as possible to the 
others. By slight changes in the assumed positions of a and b {i.e., in 
the corresponding values of 6, or of / — to), it is possible with little 



174 THE BINARY STARS 

labor to obtain a theoretical curve which fits the whole course of the 
observed curve almost as well as one determined by least squares. The 
criterion of this is that the parts of the observed curve below h (near 
totality), between a and h, and above a (near the beginning or end of 
eclipse) shall give the same mean value of k. The individual deter- 
minations of k are of very dififerent weight. Between a and h (that is 
for values of ai between 0.6 and 0.9) ^ changes very slowly with k. At 
the beginning and end of the eclipse the stellar magnitude changes very 
slowly with the time, and hence, by (13), with yp. The corresponding 
parts of the curve are therefore ill adapted to determine k. For the 
first approximation it is 'well to give the values of k derived from values 
of tti between 0.95 and. 0.99, and between 0.4 and 0.2, double weight 
(provided the corresponding parts of the curve are well fixed by obser- 
vation). The time of beginning or end of eclipse cannot be read with 
even approximate accuracy from the observed curve and should not 
be used at all in finding k. The beginning or end of totality may some- 
times be determined with fair precision, but does not deserve as much 
weight as the neighboring points on the steep part of the curve. If 
further refinement is desired, it can most easily be obtained by plotting 
the light-curve for two values of k and comparing with a plot of the 
observations. This will rarely be necessary. 

When once k is given, the determination of the light-curve is a very 
easy matter. For each tabular value of ai, equation (13) gives Bu and 
hence h — to. The values of the stellar magnitude m corresponding 
to given values of ai are already available, having been used in the 
previous work. The light-curve may thus be plotted by points in a 
few minutes. 

After a satisfactory light-curve has been computed, we may proceed 
to determine the remaining elements. Let 0'and 6" be the values 
corresponding to the beginning of eclipse (ai = o) and to the begin- 
ning of totality (ai = i). Then by (13) 

sin2 d' = A -\- Bxl/{k, o) and sin^ d" = A + B^p(k, i). 

These computed values are more accurate than those estimated from 
the free-hand curve drawn to represent the observations. At the first 
of these epochs 5 = ri + r2, and at the second d = ri — ri. We have 
then, by (10) 

n^i H- ky- = cos2 i + sin2 % sin^ B\ 
ri2(i — li? = cos2 i -f- sin2 i sin^ B", 
whence 

4/fe cot2 i= {i -ky sin2 0' - (i + yfe)2 sin^ Q\ 
4yferi2(i 4- cot2 i) = sin2 0' — sin2 Q" . 



THE BINARY STARS 175 

Introducing A and B, we have 

4k cot2 i= -4kA-{-B{(i- k)mk, o) - (I + jfe)V(*, I) }, 
4^2 cosec2 i= B [yp{k, o) - ^p{k, i)). 

The coefficients are functions of k alone, and may be tabulated. It is 
most convenient for this purpose to put the equations in the form 

- . 5 



Ti^ cosec' * 



0i(^) ' 



B 
cot^z = 77- — A, 

<f>2ik) 



(14) 



as in this way we obtain functions whose tabular differences are com- 
paratively smooth (which is not true of their rieciprocals). With the 
aid of these functions the elements may be found as soon as A and B 

are known. If - <02(^) the computed value of cot i is imaginary and 

the solution is physically impossible. It is therefore advisable to apply 
this test to the preliminary values oi A, B, and k, and, if necessary, to 
adjust them so that the solution is real. The limiting condition is 
evidently cot * = o, corresponding to central transit. 

The geometrical elements of the system are now determined. We 
are still in doubt, however, whether the principal eclipse is total or 
annular. This can be determined only by consideration of the secon- 
dary minimum. The intensities during constant phase at the two 
minima are connected by the relation ^^Xi + X2 = i. If the intensity 
at principal minimum is X^, that at the secondary minimum will be 

I — ^'^X^ if the principal eclipse is total, and if it is annular. The 

first of these expressions is always positive and less than unity. The 
second exceeds unity if i — X^ > k^. The assumption of total eclipse 
at principal minimum leads therefore in all cases to a physically pos- 
sible solution. That of an annular eclipse does so only if i — X^ is not 
greater than k^. Otherwise the computed brightness of the smaller 
star is negative. The brightness at secondary minimum will be greater 
than at the primary by i — X^(i -f k^) if the primary eclipse is total, 

and — [ I — X^(i -\-k^)] if it is annular. The latter hypothesis there- 

k^ 

fore gives rise to the shallower minimum. In many cases it may be 
impossible to decide* between the two without actual observations of 
the secondary phase. The computed depth of secondary minimum 
may, however, be so great that it is practically certain that it would 



176 THE BINARY STARS 

sometimes have been observed if it really existed. The corresponding 
hypothesis should then be rejected. If X^(i + k"^) is nearly unity, the 
primary and secondary minimum, on both hypotheses, must be of 
nearly equal depth. This can occur only if X^ < >^ ; that is, if the 
depth of minimum is less than 0.75 mag. In such a case it is probable 
that the period is really twice that so far assumed, that the two stars 
are of equal surface brightness, and that two sensibly equal eclipses 
occur during each revolution. The true values of 6 are therefore half 
those previously computed with the shorter period. If the determina- 
tion of k is repeated on this basis, and the equation X^(i -{- k^) =^ i is 
still approximately satisfied this solution may be adopted. 

Such a system presents a specialized example of [the case], when 
both primary and secondary minima have been observed and show a 

I - X2 

constant phase. In this case, by (5), k^ = where Xi corresponds 

Xi 

to the total eclipse, which, so far as we yet know, may occur at either 
minimum. As before we begin by finding from the light-curve the 
values of sin^ d corresponding to given values of ai. From a few of 
these, by the method already described, an approximate value of k may 
be obtained which is sufficient to show which of the values given by 
(5) on the two possible hypotheses is the correct one. 

We have next to find the light-curve which gives the best represen- 
tation of the observations consistent with the value of k given by (5). 
The form of the light-curve now depends only on the constants A and 
B in the equation 

sin2 $1 = A -\- BrPik, ai). (13) 

Approximate values of these constants may be derived as above from 
the values of sin'^ 6 when a = 0.6 and 0.9. These may be improved 
by trial and error, which will be aided by plotting the resulting light- 
curves along with the observations, and, if the data warrant it, may 
finally be corrected by least squares. When satisfactory values of 
A and B have been determined, the final light-curve may be computed 
by (13), and the elements by (14), as [before], except that here there 
is no uncertainty as regards the nature of the principal eclipse. 

In review of the foregoing, it may be remarked that the method of 
solution is direct and simple. It involves a very moderate amount of 
numerical work, of which the greater part — namely, the determina- 
tion of the values of the magnitude, time, and .position in orbit (0) 
corresponding to different percentages of obscuration (a) — requires no 
modification during the successive approximations. The light-curve 



THE BINARY STARS 177 

may be found without the necessity of computing the elements, and 
with two or three trials may be determined so as to represent the whole 
course of the observations, making the laborious solution by least 
squares superfluous except in the case of observations of unusual pre- 
cision. Such a solution itself is much simplified if the constants de- 
fining the light-curve, instead of the elements of the system, are treated 
as the fundamental unknowns, as the coefficients of the equations of 
condition may be easily found graphically with the aid of data already 
computed. The elements may be found, at any stage of the process, 
by a few moments' calculation, from the constants defining the light- 
curve. 

Russell's paper contains a number of tables, of which the 
four directly applicable to the cases of total or annular eclipses 
which have been discussed are reprinted here. His Table I, 
tabulating the function given in equation (9), is omitted, 
though it is fundamental, because it is used, so far as we are 
at present concerned, only in constructing the subsequent 
tables. 

Table II contains the function yf/ik, ai) defined by the equation 

{l-hkp{k,ar)Y-{l + kp(k,a,)Y 
'^^^' '''^ ~ jl + kp{k,a,)Y-{l + kplik,a^)y 

(where a^ = 0.6 and as = 0.9), which is used in determining k in the 
case of total eclipse. The uncertainty of the tabular quantities does 
not exceed one or two units of the last decimal place, except for the 
larger values of xf/, corresponding to values of ai less than 0.3, for which 
the actual errors may be greater, but are not more serious in propor- 
tion to the whole quantity tabulated. 
Table I la contains the functions 

*'^*^ =^(*,o)-^(*,i)'"'^ *'^*' =(i-«V(*,o)-(. + *)V(*..),' 

which are useful in determining the elements in the case of total eclipse. 
Table A gives the loss of light (i - X), corresponding to a given 
change Aw in stellar magnitude. For a difference of magnitude greater 
than 2.5, the loss of light is 0.9000+ one-tenth of the tabular value for 
Aw — 2™5. Table B gives the values oi 6 — sin d for every o.oi of 
(expressed in circular measure), and saves much labor in computing 
the values of sin d corresponding to a given interval from minimum. 



178 



THE BINARY STARS 





o o 


ON 


fe «s »o 


o 


^ 


NO 


Tf 


re 


fs 


^ 


^ 


<N 


NO to N 


00 


^ 


NO 


I-I 




vO 


^ 


Ov g 


r< 


9 ^ ^ 


t^ 


HH 


m 


o 


lO 


o 


m 


Q 


m 


O NO re 


o 


o 


r< 


OS 


r^ 


ON 


00 


so rJ- <N 


o 


ON 


t>- 


so 


"^ 


re 


l-t 


o 


>-l 


re >* NO 


00 


o 


w 


re 


Tf 


to 


o 


+ 




+ " " 


+ 


o 




o 

+ 






o 

+ 


o 


o 

1 


o 

1 1 1 


o 

1 


T 


r 


T 


T 


T 




"t 'O 


l^ 


^ l^ SO 


so 


00 


t^ 


re 


re 


■rf 


so 


^ 


sn 


■<:*■ lO On 


(N 


^ 


Tt 


OS 


^^1 


o 


lO i-i 




»o ce re 


rh so 


On 


re 


r^ 




lO 


Q 


lO 


HH t^ re 




o 




NO 




lO 


Tl- CS 


o 


l^ lO re 




ON 


1^ 


so 


rf 


re 




o 




re -^ so 


00 


i 


r« 


re 


to 


o 


+ 


N 


+ " " 


+ 


o 


o 


o 

+ 






o 

+ 


o 


o 

1 


o 

1 1 1 


o 

1 


T 


T 


T 


T 


T 




lO 00 


^ 


rf M re 


<N 


OS 


r^ 


»o 


^ 


f* 


Tf 


„ 


o 


O re t^ 


00 


„ 


re 


o 


On 


to 


% 


lo r^ 


Tf 


re 00 »o 


rf 


re 


ITi 


t^ 


o 


re 


so 


B 


so 


(N 00 rj- 




f*^ 


o 


to 




n 


t^ "-t 


CS| 


ON so -^ 


M 


o 


00 


so 


lO 


re 






re Tj- so 


00 


^ 


c< 


re 


"^ 


to 


6 




c< 


+ " "^ 


-f 


■ 


o 


o 






o 

+ 


o 


o 

1 


o 

1 1 1 


o 

1 


T 


T 


T 


T 


T 




«+ 00 


OO 


►H >-l ►-! 


"+ 


\r> 


„ 


^ 


o 


00 


1-1 


o 


r^ 


•1 -^ OS 


lO 


^ 


o 


ON 


o 


r1- 


o 


O vO 


00 


re 'I- 00 


«+ 




l_l 


<N 


re 


rf 


r^ 


NO 


re OS lO 


fS 


g 


ON 


<N 


ON 00 1 


•-. t^ 


-^ 


►-1 00 lO 


re 


t-l 


ON 


t^ 


lO 


re 




o 




re Tt so 


00 




re 


re 


■^ 


6 


+ 


M 


M I-I I-I 

+ 


1-1 

+ 


" 


o 


o 






o 

+ 


o 


o 

1 


o 

1 1 1 


o 

1 


T 


T 


T 


T 


T 




^^ 


r^ 


00 '^- so 


re 


»o 


<N 


o 


so 


o 


„ 


„ 


Tf 


Th 00 O 


re 


o 


On 


on 


<N» 


to 


o 


t>. 


ID M r< 


lO 


o 


00 


r^ 


so 


r^ 


00 


g 


t^ 


^ O t^ 


re 


»^ 


o 


NO 


"^ 


ID H* 


t^ 


re O t^ 


Tf 


(S 


ON 


!>. 


m 


re 




n 


re »o so 


00 


o 




re 


re 


rt- 


d 




(s» 


M C) i-i 

+ 


-f 


" 


o 


o 
4- 






o 


o 


o 

1 


o 

1 1 1 


o 

1 


T 


T 


T 


T 


T 




't '^ 00 


I^ O 00 


^ 


^ 


^_ 


lO 


re 


<N 


„ 


^ 


^ 


Tl- (N) rh 


re 


^ 


so 


-& 00 


M 


O 


00 o 


hH 


n Tf ON 


ON 




so 


n 


o 


ON 


ON 


Q 


00 


lO <S 00 


Tt 


8 


NO 


00 


rs» 


o 


OS lO 


HH 


so (N 00 


lO 


re 


o 


00 


so 


re 




o 




re «o so 


00 




<N 


re 


^ 


d 


+ 


fC 


r< (N 11 
+ 


+ 






o 

+ 






o 

+ 


o 


o 

1 


o 

1 1 1 


o 

1 


T 


r 


T 


T 


r 




vO O 


Tt- 


O ^ O 


o 


re 


r^ 


rt- 


ON 00 


M 


Q 


_ 


ON ON O 

so re o 


rh 


^ 


(N 


so 


re 


■rf 


«^ 


lO Q 


rO 


so O i-i 


so 


Tf 


«o 


ON 


■^ 


)_l 


o 


Q 


On 


to 


g 


to 


to 


On 


to 


lo 5 


lO 


ON lO HH 


t^ 


rf 




00 


so 


^ 


n 


6 




re »o t^ 


00 




(S 


M 


re 


6 


^ "* 


re 


fS C< P< 


M 


M 


hH 


o 






o 


o 




o 


o 


HH 


t_l 


l_l 


hH 


H4 


+ 




+ 


+ 






+ 






+ 




1 


1 1 1 


1 


1 


1 


1 


1 


1 




J^^ 


r^ 


'd- so ■+ 


ON 


re so 


OO 


lO 


^ 


t^ 


„ 


<N 


r^ 00 t^ 


re 


^ 


■Ti- 


o 


o 


r^ 




-+ 


SO fN t^ 


so 


o 


r>. 


r^ 


o 


lO 




g 


o 


00 »o I-I 


so 


^ 


re 


(N 


to 


ON 




(S vO 


o 


re 00 re 


ON so 


M 


ON 


t^ 


"^- 


c< 


n 


re lo r^ 


00 


i 




rj 


C4 


cs 


6 


ID r*- 


Tt 


re M <N 


l_ 


M 


1^ 


o 






o 


o 


o 


o 


o 


l_l 


^ 


|_4 


|_4 


H^ 


+ 




+ 


+ 






+ 






+ 




1 


1 1 1 


1 


1 


1 


1 


1 


1 




8 f:^ 


-t 


lO 00 fS 


<N 


re 


iO 


c5^ 


r^ 


^ 


re 


^ 


Tt- 


00 Tt- 00 


1-- 


f, 


(N 


NO 


ON 


^ 




o 


OS ■<1- I-I 


re 


o 


<N 


r>. 


as 


re 


8 




O 00 re 


t^ 


Q 






ON 


re 




<s rn 


t^ 


00 <N 1^ 


M 


00 


'+ 


o 


t^ 


-t 


<s 


M 


Ti- lo r^ 


00 


o 


t-l 


•-I 




CSI 


6 


vO »0 


Tf- 


re re <N 


M 


l_l 


HH 


HH 


o 




o 


o 


o 


o 


o 


)-l 


HH 


I-I 


I-I 


HH 


+ 




+ 


+ 






+ 






+ 




1 


1 1 1 


1 


1 


1 


1 


1 


1 




00 r>. so 


lO On -^ 


g 


00 


J_ 


lO 


^ 


lO 


00 


„ 


^ 


lO re lO 


re 


^ 


to 


NO 


ON 


to 




t^ lO 


l_t 


<N re OO 


00 


Tt- 


Tf 00 


U-5 


lO 


o 


re 


re 11 NO 


On 


o 


00 


<N 


re 


to 


0\ 


Tf Tl- O 


so 00 •-• 


so 


o 


NO 


C) 


00 


ID 


c< 


o 


r< 


Tt- so t^ 


00 


(J 


o 








6 


t>. SO 


lO 


rf re re 


rs 


n 


l_l 


l_l 


o 




o 


o 


o 


o 


o 


HH 


l_l 


HH 


tH 


H4 


4- 




+ 


+ 






4- 






+ 




1 


1 1 1 


1 


1 


1 


1 


1 


1 




































to 


re 


o 




Tj- \n 


M 


ON lO so 


00 


rj 


OS 


o 


o 


00 


8 


o 


00 


o o »o 


M 


o 


to 


(N 


"I- 


to 




-^J 


"I- 


lO lO o 


m 


n 


r^ 


ON 


^ 


-+ 


8 


lO 


00 so O 


M 


o 


^ 


so 


NO 


sO 




o 


t^ t^ ON 




lO 


ON 


^ 


o 


NO 


re 


(S 


■^ NO 00 


OS 


o 


o 


o 


o 


II 


ON 00 


t^ 


ID n- re 


re 


(N 


H4 




)_l 


o 


o 


o 




o 


o 


HH 


hH 


i-l 


l_l 


HI 


+ 




+ 


+ 






+ 






+ 




1 


1 1 1 


1 


1 


1 


1 


1 


1 




o n 


lO 


O lO o 


ir> 


o 


lO 


o 


lO 


o 


lO 


o 


lO 


o to o 


to 


o 


to 


00 


OS 


8 


ct 


o o 


o 


l-< hH d 


M 


re 


re 


■^ 


rl- 


lO 


lO so 


so 


i-* r^ 00 


00 


OS 


ON 


ON 


ON 


o 




o 


o 






o 






o 






o 


o 






o 




l-H 



THE BINARY STARS 

TABLE IIo 
For Computing the Elements in the Case of Total Eclipse 



179 



k 


Mk) 


<t>t{k) 


1. 00 


0.380 


0.939 


0.95 


.401 


.894 


.90 


.417 


.848 


0.85 


0.427 


0.802 


.80 


431 


.755 


•75 


431 


.709 


0.70 


0.427 


0.663 


.65 


.419 


.617 


.60 


.406 


.572 


0.55 


0.390 


0.527 


.50 


.371 


.482 


45 


.349 


436 


0.40 


0.323 


0.390 


.35 


.294 


.345 


•30 


.262 


.298 


0.25 


0.226 


0.250 


.20 


.187 


.202 


•15 


.145 


.153 


O.IO 


O.IOO 


0.103 


.05 


.052 


.052 


.00 


.000 


.000 



i8o 



THE BINARY STARS 



TABLE A 

Loss of Light Corresponding to an Increase Am in Stellar Magnitude 



Am 





I 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 


o.oooo 


0.0092 


0.0183 


0.0273 


0.0362 


0.0450 


0.0538 


0.0624 


0.0710 


0.0795 


.1 


.0880 


.0964 


.1046 


.1128 


.1210 


.1290 


.1370 


.1449 


.1528 


.1605 


.2 


.1682 


.1759 


.1834 


.1909 


.1983 


.2057 


.2130 


.2202 


.2273 


.2344 


•3 


.2414 


.2484 


.2553 


.2621 


.2689 


.2756 


.2822 


.2888 


.2953 


.3018 


4 


.3082 


.3145 


.3208 


.3270 


.3332 


.3393 


.3454 


.3514 


.3573 


.3632 


•5 


0.3690 


0.3748 


0.3806 


0.3862 


0.3919 


0.3974 


0.4030 


0.4084 


0.4139 


0.4192 


.6 


.4246 


.4298 


.4351 


.4402 


.4454 


.4505 


.4555 


.4605 


.4654 


.4703 


•7 


•4752 


.4848 


.4848 


.4895 


.4942 


.4988 


.5034 


.5080 


.5125 


.5169 


.8 


.5214 


.5258 


.5301 


•5344 


.5387 


.5429 


.5471 


.5513 


.5554 


.5594 


•9 


.5635 


.5675 


.5715 


.5754 


.5793 


.5831 


.5870 


.5907 


.5945 


■5982 


I.O 


0.6019 


0.6055 


0.6092 


0.6127 


0.6163 


0.6198 


0.6233 


0.6267 


0.6302 


0.6336 


I.I 


.6369 


.6403 


■6435 


.6468 


.6501 


.6533 


.6564 


.6596 


.6627 


.6658 


1.2 


.6689 


.6719 


.6749 


.6779 


.6808 


.6838 


.6867 


.6895 


.6924 


.6952 


1.3 


.6980 


.7008 


•7035 


.7062 


.7089 


.7116 


.7142 


.7169 


.7195 


.7220 


1.4 


.7246 


.7271 


.7296 


.7321 


.7345 


.7370 


.7394 


.7418 


.7441 


•7465 


1-5 


0.7488 


0.751 1 


0.7534 


0.7557 


0.7579 


0.7601 


0.7623 


0.7645 


0.7667 


0.7688 


1.6 


.7709 


.7730 


.7751 


.7772 


.7792 


.7812 


.7832 


.7852 


.7872 


.7891 


1-7 


.7911 


.7930 


.7949 


.7968 


.7986 


.8005 


.8023 


.8041 


.8059 


.8077 


1.8 


.8095 


.8112 


.8129 


.8146 


.8163 


.8180 


.8197 


.8214 


.8230 


.8246 


1.9 


.8262 


.8278 


.8294 


.8310 


.8325 


.8340 


.8356 


.8371 


.8386 


.8400 


2.0 


0.8415 


0.8430 


0.8444 


0.8458 


0.8472 


0.8486 


0.8500 


0.8514 


0.8528 


0.8541 


2.1 


.8555 


.8568 


.8581 


.8594 


.8607 


.8620 


.8632 


.8645 


.8657 


.8670 


2.2 


.8682 


.8694 


.8706 


.8718 


.8729 


.8741 


.8753 


.8764 


.8775 


.8787 


2.3 


.8798 


.8809 


.8820 


.8831 


.8841 


.8852 


.8862 


.8873 


.8883 


.8893 


2.4 


.8904 


.8914 


.8924 


.8933 


.8943 


.8953 


.8962 


.8972 


.8981 


.8991 


2.5 


0.9000 


0.9009 


0.9018 


0.9027 


0.9036 


0.9045 


0.9054 


0.9062 


0.9071 


0.9080 



For values of Am greater than 2.5, the loss of light is 0.9000 plus j^ of 
the loss of light corresponding to Am — 2.5. 



i 



THE BINARY STARS 

TABLE B 

Values of d — Sin 6 



i8i 





0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


0.00 


0.0000 


0.0002 


0.0013 


0.0045 


0.0105 


0.0206 


0.0354 


0.0558 


0.0826 


0.1 167 


.01 


.0000 


.0002 


.0015 


.0049 


.0114 


.0218 


.0372 


.0582 


.0857 


.1205 


.02 


.0000 


.0003 


.0018 


.0055 


.0122 


.0231 


.0390 


.0607 


.0888 


.1243 


.03 


.0000 


.0004 


.0020 


.0060 


.0131 


.0244 


.0409 


.0632 


.0920 


.1283 


.04 


.0000 


.0005 


.0023 


.0066 


.0141 


.0258 


.0428 


.0658 


.0953 


.1324 


0.05 


0.0000 


0.0006 


0.0026 


0.0071 


0.0151 


0.0273 


0.0448 


0.0684 


0.0987 


0.1365 


.06 


.0000 


.0007 


.0029 


.0078 


.0161 


.0288 


.0469 


.0711 


.1022 


.1407 


.07 


.0001 


.0008 


.0033 


.0084 


.0171 


.0304 


.0490 


.0739 


.1057 


.1450 


.08 


.0001 


.0010 


.0037 


.0091 


.0183 


.0320 


.0512 


.0767 


.1093 


.1494 


.09 


.0001 


.0011 


.0041 


.0098 


.0194 


.0337 


.0535 


.0796 


.1130 


.1539 



To illustrate Russell's method I have chosen his orbit of 
W Delphini, which is a "typical Algol variable with a deep 
primary minimum, showing a constant phase, and little or no 



9'?o 



1090 



u^o 



jaiPo 



--X 


V, 






^ 


,-- 




\ 






/ 








\ 












^^ n a 













-o?3 -o?2 -o4j o?o +o?i +o*a o'?9 

Figure ii. Light-Curve of the Principal Minimum of W Delphini 



1 82 



THE BINARY STARS 



secondary minimum." Its light curve, "defined by the 500 
observations by Professor Wendell, with a polarizing photo- 
meter, which are published in the Harvard Annals, 69, Part i," 

TABLE a 
Observed Magnitudes 



Phase 


Mag. 


No. 
Obs. 


0.- 


-C. 


Phase 


Mag. 


No. 
Obs. 


O.-C. 


-0^2894 


9.41 


6 


+ 


"01 


+ 0^0560 


11.76 


7 


^-o^^oi 


.2637 


9.49 


5 


+ 


.02 


.0659 


11.58 


8 


+ .01 


.2458 


9.58 


5 


+ 


.04 


.0753 


11-33 


7 


- .04 


.2306 


9.59 


4 


- 


.01 


.0859 


II. 14 


5 


— .02 


.2200 


9.67 


5 




.00 


.0937 


10.97 


5 


- .05 


.2106 


9.73 


8 


+ 


.01 


.1036 


10.88 


8 


+ .02 


.2007 


9.79 


10 




.00 


.1147 


10.73 


8 


+ .05 


.1911 


9.88 


12 


+ 


.02 


.1246 


10.56 


12 


+ .03 


.1817 


9-95 


10 


+ 


.01 


.1351 


10.39 


14 


.00 


.1718 


10.02 


8 




.00 


.1445 


10.31 


II 


-f .04 


.1615 


10.16 


17 


+ 


.04 


.1546 


10.13 


10 


— .02 


.1506 


10.23 


14 




.00 


.1641 


10.10 


II 


+ .04 


.1396 


10.37 


14 


+ 


.01 


.1744 


9.97 


10 


.00 


.1311 


10.44 


16 


— 


.03 


.1847 


9.90 


9 


+ .02 


.1212 


10.59 


17 


— 


.03 


.1941 


9-79 


9 


— .02 


.1121 


10.78 


14 


+ 


.01 


.2050 


9.71 


8 


— .02 


.1013 


10.91 


17 


— 


.04 


.2157 


9.71 


6 


+ .04 


.0906 


II. 12 


14 


- 


.01 


.2242 


9.63 


8 


+ .01 


.0809 


11.30 


10 


— 


.02 


.2345 


9.57 


7 


.00 


.0715 


11.51 


12 




.00 


.2507 


950 


7 


.00 


.0617 


11.69 


10 




.00 


.2708 


9.48 


7 


+ .03 


.0509 


11.88 


7 




.00 


.2811 


9.43 


4 


+ .02 


.0313 


12.05 


5 


— 


.04 


.94 


9.42 


5 


4- .02 


.0169 


12.08 


4 


— 


.02 


1.90 


9-35 


5 


- .05 


— .0082 


12.07 


7 


— 


.03 


2.04 


9.41 


7 


+ .01 


+ .0060 


12.16 


5 


+ 


.06 


2.67 


9.38 


5 


— .02 


.0139 


12.09 


4 


— 


.01 


3.04 


9.42 


3 


+ .02 


.0261 


12.03 


5 


- 


.07 


4.04 


9-44 


6 


+ .04 


.0356 


12.02 


6 


— 


.03 


4.48 


936 


7 


- .04 


+ .0460 


11.87 


6 


- 


.04 











is shown in Figure 1 1 . The observations have been combined 
into the normal places given in Table a, on the basis of a 
period of 4.8061 days, which was found to require no correction. 



THE BINARY STARS 183 

From the thirty-eight observations outside minimum we find the 
magnitude during constant light to be 9^395 =t 0.009. There is no 
evidence of any change during this period. With a circular orbit, the 
secondary minimum should occur at phase 2^40. As none of the ob- 
servations fall within 0^27 of this, they give us no information whether 
such a minimum exists. The light-curve of the principal minimum is 
very well determined. The eclipse lasts from about — 0^28 to + 0^28, 
and there is a short constant period at the middle, of apparently a little 
less than one-tenth the total duration of the eclipse. The mean of the 
twenty observations lying within 0^02 of the middle of eclipse gives 
for the magnitude at this phase 12^10 ±0.014. The range of variation 
is therefore 2']'70, and the light-intensity at minimum 0.0832 times 
that at maximum. This shows at once that the eclipse is total, for if 
it was annular, the companion (even if perfectly dark) must cut off at 
least 0.917 of the light of the primary, and hence its radius cannot be 
less than 0.956 times that of the latter. In such a system the duration 

0.044 

of the annular phase could not exceed , or 0.022 of the whole 

1.956 

duration of eclipse. The observed constant phase is almost five times 

as long as this. 

The brighter star therefore gives 0.9168 of the whole light of the 

system, and if isolated would appear of magnitude 9.49; while the 

fainter but larger star which eclipses it gives out only one-eleventh as 

much light, and when seen alone at minimum is of magnitude 12.10. 

The loss of light (i — 1) at any given time, /, will be o.9i68ai, 
since ai is the percentage of obscuration. For a series of values 
of ai we tabulate the values of (i— 1) and then take from 
Table A the corresponding changes of magnitude and apply 
them to magnitude 9.4. Next, from the free-hand curve 
drawn to represent the data of observation, we read ofT the 
epochs ti and /2 at which the magnitudes so computed are 
reached before and after the middle of eclipse. Half the 
difTerence of /i and ^2 may be taken as the interval / from the 
middle of eclipse to each phase and the corresponding value of 
d formed from 

2 TT 

6 = / = 1.3065^, where d is expressed in radians and t in days. 

With the aid of Table B sin 6 is found and then sin^ 6. These 
quantities are all entered in Table b. 



1 84 



THE BINARY STARS 



.• Tf lO O M •• 

VO O O 00 'O vO 

lO lO lO Tt- tJ- to 

6 






•• »0 00 •• 
00 N M O 
lO lO lO »o 



00 O (N •-< 
00 Th O ON O ^ 
(S vO On N 00 rc 



+ 



o o 

+ 



NO "^ O 

t^ ON Q 
rO NO O 

do-: 

I I I 



lO t^ 00 -^ 

lO t^ w o 

•-« N CO Th 



I I I I 



lO ^ 


l-H to t^ 


ro C 


> r^ 




On 


O (N 


O 00 


ON ro O 


ON < 


> ON 


t>. lO On 


<N| 


rj- NO 


= « 


Tj- ro fN« 




^ 5^ 


•-1 r« ci 


ro 


fO CO 


o o o 


q d 


O O O 


O 


q q 



I I I I I I I 



rl- O O 


(N 


NO f^ ON 


r< 


o 


^^ 


HH O 


O^ t^ 


t^ »0 NO 


O 


1^ NO NO 


r^ 


On 


1-1 


t>> •^ 


n o 


? ^ q 


S- 


iP ^ S? 

o o o 


q 


q 


q 


8 8 


8. 8 



.•Ttoo•-«NO^Ol-«»-'•-lT^NOC^NO 
•^OnOOO CS t>.fONO00 lOfOrOfONO 

ONtOONNOrJ-tHONNO ~ ~ 

rOfOC<NP*N>-<M 

6 * 



fO O 00 vO lO C< 

"-I •-> o o o o 



.. lOOO O O -^O Tt-0000 

8N- iOfO<Orhr^l^Tj-oo 
lori ooono t1-(n o 
rO M <N n 1-1 f 



n O t^ NO tJ- ro "-I 

►-I 1-1 o o o o o 



• • o mioThCNjNO o o Ti-ioloo •• 
■"^Thoo 1-^000000 t^r^M too fO>-i 

O lO(N).OOONO Tj-(NJ OoonO lOThM 

rONM<N'-'»-i'-i'-i'-iOOOOO 
o 

I 1 



8 100 OvvOnOnO ThiDM ThTt-«00 
O <Nj tJ-OnnOnO >- rOONM OOOO O 
TfiONO t^OO O CNJ »O00 MnOOO On*-" 

S'dNONONONd666i-:>-J«>-:r< 

On i-ii-ii-thHi-ii-ii-ii-iMi 



r^-^o t^rho t>.Tj-o 

1-1 rOiOvOOO O M r0»0 

OnoO t^NO lOiOrJ-rorj t^ON© 

O "-i M fO-^iONO 1^000000 O^ O-^ 



ON lO vO 00 
O 00 t^ NO 



O « 

d 



»0 NO 1^ 00 



be 
.S "^ 

i ^ 



-I 

60 a 
o rt 



.t: a 



a o 

<v a 
05 B 






1^ 

E rt 



3 :5 
« 6 



THE BINARY STARS 1 85 

From the values of ti and <2 it appears that the observed curve is 
remarkably symmetrical, and that the actual epoch of mid-eclipse is 
0.0015 days earlier than that assumed by Wendell. The times of 
beginning and ending of the eclipse cannot be read accurately from 
the curve and are marked with colons to denote uncertainty. 

From the values of sin^ 6 we have now to find k with the 
aid of Table II. From equation (13) we have 

sin' 9i — A 



^(*.ai) = 



B 



hence, if we let A be the value of sin^ 6 when ai = 0.6 and A — B 
its value when ^2 = 0.9, we may find a value of k for every 
tabulated value of ai by inverse interpolation in Table II. 
Thus, taking A = 0.0369 and B ( = sin^ 62 — sin^ ^3) = 0.0258, 
as given by our curve, we find for ai = 0.0 that \f/ {k, ai) = 
+ 4.28:, and hence, from the first line of Table //, k = 0.56:. 
Colons are here used because the values of k are less accurate 
when the tabular differences of \f/ {k, ai) are small. 

The values of k are seen to be fairly accordant except for 
those corresponding to values ai near 0.6. Inspection of 
Table II "shows that this discrepancy may be almost removed 
by increasing all the values of xj/ by 0.024 — which may be 
done by diminishing A by 0.024^. Our new value of A is 
therefore 0.0363." The new set of ^'s are found to be dis- 
cordant for values of ai near 0.9; "but by diminishing B by 
2}^ per cent, [giving B = 0.0252], and hence increasing all 
the computed values of xf/ in the corresponding ratio, we obtain 
a third approximation of a very satisfactory character." The 
general mean is now k = 0.528. 

With these final constants, A = 0.0363, B = 0.0252, k = 
0.528, we may compute a theoretical light-curve and also the 
elements of the system from equation (14). Table c gives the 
second and third approximations to the value of k and the 
data for the final light-curve. 

Plotting the magnitudes computed in Table a against the epochs 
— 0*10015 ±/, we obtain the computed light-curve. The residuals 
(O. — C.) are given in the last column of Table a. Their average value, 
regardless of sign, is 0^020. 



1 86 



THE BINARY STARS 









r< M 


t^ »0 lO VO 


VO 


t^ VO 




•* Ov to 00 








On lO 


(N O 00 VO 


rt- 


« 2 


oo 


vO Tt- rh M 

q q q q 






■^ 


(S (M 


rj M M i-< 






q 








6 
















^ 


ro "«*• 


M ID O lO 


HH 


lO 00 


xn 


Th rh VO VO 








r>. (s 


On VO Tf M 


Ov VO to 


o 


00 VO >o to 






c 


CO fO 


M M n CI 




hH l-H 




q q q q 






w 


d d 


. 












a 


















% 





















































u 


^ 


IC vO 


tJ- i-i lO Th 


to 


to o 


hH 


O t-i 1-1 to 








ON ■* 


ir> o t^ VO 


VO 


t^ Ov 


HH 


t^ -^ to i-i 






.S 


to o 


^ s-?? 


q 


q q 


q 


q q S q 




o 


'» 




















o o 














J 


















h) 



















< 


















z 


















fe 




CS ro 


t-i oo H >-i 


Q 


O to 


r« 


to (v» (S O 




■^ 


to 00 


Ov to "-I O 


g 


Ov t^ 


lO 


Ov n to lo 






2 ^ 


rt- to n HH 


o ^ 


n 


M to to to 






tt) 


q q q q 


o 


q q 


q 


q q o o 








d 






1 1 


1 


till 






^_^ 




































ti 


O rO 


^^^8 


Q 


00 Ov 


Q 


(M VO 00 Ov 






. 


O ►-, 


Q 


lO 00 


8 


VO 1^ HH 00 






00 


1-1 r^ 


Ov to 00 rt- 


o 


to VO 


HH n to to 






lO 


4 ci 


H^ M d d 


d 


d d 


t_^ 


M 1-4 M 1^ 






3 


+ 


+ 




1 1 


1 


1 1 1 1 






^ 




















•• "^ 00 1^ M •• 








Th 1^ M .. 








00 to 


to M to "-I 




Tf hH 




VO O HH 00 






Jii 


lO lO 


ID lO »0 ID 




lO lO 




lO lO lO -^ 




X 




d 


. 








d 




o 


















s? 


















CLi 


















< 






















Tf rh ir> rt- 


lO 


O lO 


Q 


1^ M rh <v) 




Q 




d rO 


l^ -rh 'i- Ov 


(V| 


\0 00 


§ 


lO 00 M i-i 




ro 


-^ 


Tj- I^ 


Ov to 00 to 


o 


to VO 


•I n to T*- 






-4- cl 


>^ .:, 6 6 


d 


d d 


HH 


HH hH •-( 1-^ 








+ 




+ 


1 1 


1 


1 1 1 1 








.. C) 


H^ O O 1^ 








rf- O »^ .. 








\D >-, 




00 O 




M HH ON in 






-Oi 


d 


lO lO lO '^ 




't ^ 




t^ VO »0 lO 

' ■ ' d 




>< 


















O 

a 

Oh 
Ok 




.. lO 


W Tf- VO «0 


■<t 


n o 


VO 


•-I ro tJ- .. 






O VO 


to •- (N 00 


(N 


to r^ 


l^ 


tO lO Ov 00 




-^ 


fO VO 


Ov to 00 to 


o 


to VO 


Ov 


*>^ CJ cj to 




< 


4 pi 


h; M d d 


d 


d d 


d 


i4 )-? K-i i-i 









+ 




+ 


1 1 


1 


Mil 




N 
































to 00 ov O 

Ov qv Ov q 






a 


O w 


W to ri- lO 


vq 


^ 00 


as 








d 













THE BINARY STARS 187 

From Table I la we find for k = 0.528, tt>\{k) = 0.382, 4>2{k) = 
0.507; whence 

B 

^ot^* = ~7i; - ^ = 0-0133. cot t = O.I 15, t = 83° 25'. 

B 
r^ cosec^ i = — — = 0.0660, r^ = 0.0652, ri = 0.256, 

and finally 

r% = ^ri = 0.135. 

In other words, we have, taking the radius of the orbit as unity, 

Radius of larger star 0.256 

Radius of smaller star * ' 0.135 

Inclination of orbit plane 83 "25' 

Least apparent distance of centers 0.114 

Light of larger star 0.0832 

Light of smaller star 0.9168 

[Period of revolution 4.8061 days] 

At the middle of eclipse, the larger star overlaps the other by only 
0.007 of the radius of the orbit, or about one-twentieth of the radius 
of the smaller body, so that the eclipse is very nearly grazing. The 
smaller star gives off eleven times as much light as the other, and ex- 
ceeds it forty-fold in surface-brightness. 

The loss of light at secondary minimum should be k times the light 
of the fainter star, or 0.023 of that of the system. The corresponding 
change in stellar magnitude is 0.027, which could only be detected by 
refined observations. 

In an earlier paper Russell gives the equations which deter- 
mine the density of the components relatively to the density 
of the Sun in terms of the orbit elements. 

Let the total mass of the system be m, that of the larger star my 
and that of the smaller m{\ — y). If a is the semi-axis major of the 
orbit, we shall have a = KwYi P^i, where X is a constant depending 
on the units of measurement. If we choose the Sun's mass, the Sun's 
radius, and the day as units, then for the Earth's orbital motion 
a = 214.9, P = 365.24, whence K = 4.206. 

In determining the elements of the system, we have taken a as our 
unit of length. The actual radius of the larger star is therefore an, 
and its volume in terms of that of the Sun is K^mP^ri^, or ^44 mp^ri\ 



l88 THE BINARY STARS 

y 

Its density pi is therefore pi = 0.01344 > and similarly, of the 

I — y. 
smaller star p2 = 0.01344 . 

The actual densities can be computed only when the ratio 
of masses of the two stars is known. Assuming the two com- 
ponents of W Delphini to be of equal mass, the equations give 




r-© 



o 

SUN 




:--© 



Figure 12. The Sj^stem of W Delphini. The relative orbits of the bright star 
are shown; the upper diagram representing the elements as given in the 
accompanying solution, the lower, Shapley's elements on the assumption that 
the stars are darkened to zero at the limb. The diameters of the disks of 
stars and Sun are drawn on the same scale; the three bodies have the same 
mass, but the stars are less dense than the Sun. — From Shapley's article in 
Popular Astronomy, vol. 20, p. 572, 1912 

O.I 18 of the Sun's density for the density of the smaller and 
brighter star and 0.017 ^oi" the density of the larger and 
fainter one. 

Dr. Harlow Shapley, who was associated with Professor 
Russell in the theoretical investigations which have been 
described above, has applied the theory in a very comprehen- 
sive manner, and has published (in No. 3 of the Contributions 
from the Princeton University Observatory, 191 5) "A Study of 
the Orbits of Eclipsing Binaries" containing orbits for ninety 
different systems, with a thorough discussion of the results. 
It is not expedient to copy his table of orbits here, for, aside 
from the space required, it would be necessary to extend con- 
siderably the discussion already given, in order to make all of 



THE BINARY STARS 189 

the details of the table readily intelligible. Some general 
statements concerning the table and Shapley's discussion must 
suffice. 

Since there is both theoretical and observational support 
for the hypothesis that the disks of eclipsing variable stars 
are not uniformly illuminated but are darkened toward the 
limb, like our Sun, by the increased absorption of their en- 
veloping atmospheres, Shapley gives for each system at least 
two orbits, one for the hypothesis of uniform illumination, the 
other for the hypothesis of darkening toward the limb. When 
the nature of the data does not permit a definitive solution, 
several orbits are given to show the limits within which the 
elements may be varied. In all, more than 200 orbits are 
tabulated, nearly all of them computed by Shapley himself. 
The results seem to justify the conclusion that darkening 
toward the limb characterizes practically all of the stars under 
discussion, but the degree of darkening cannot as yet be deter- 
mined with any certainty. 

Shapley finds a distinct correlation between the range of 
light variation and the relative sizes of the two components; 
when at principal minimum the star is two magnitudes or more 
fainter than when at maximum, the brighter star is never the 
larger, and it is only rarely the larger when this range is in 
excess of one magnitude. On the other hand, when the range 
is less than 0.7 magnitudes, "there is not a single system 
known where the fainter star is the larger." 

There is a positive indication in all but a very few cases 
"that the fainter star is self-luminous and in no case is it nec- 
essary to assume one component completely black. In about 
two-thirds of the systems the difference in brightness of the 
components does not exceed two magnitudes, and no observed 
difference is greater than four magnitudes." A large percen- 
tage of the visual binaries have a far greater difference of 
brightness between the two components. The fainter star, in 
the eclipsing systems, has always been found to be redder, and 
hence probably of a later class of spectrum than the bright 
star, whenever it has been possible to determine the relative 
color-index of the two components. 



190 THE BINARY STARS 

The great majority of the ecHpsIng binaries at present 
known belong to the spectral classes B and A. When the den- 
sities are computed, on the assumption that the mass of each 
component is equal to that of the Sun, it is found that the 
densities of stars of spectral classes B and A lie mainly between 
0.02 and 0.20 of the density of the Sun. The range in density 
of the small number of eclipsing binaries of spectral classes 
F, G, and K is much greater, exceeding the density of the Sun 
in one system (W Ursae Majoris, Class G) and falling below 
o.oooi of the Sun's density in two instances, while in only two 
does it fall within the limits 0.02 and 0.20. 

Definite values for the eccentricity of the orbit could be 
determined in only about a dozen systems, the maximum value 
being 0.138 (for R Canis Majoris); a number of systems are 
known to be practically circular, but in most systems the evi- 
dence on this point is insufficient. However it may safely be 
said that the departure from a circular orbit is never very great. 

Whenever there was evidence of ellipticity of the disks of the 
two components, caused by their mutual attraction, it was 
found that "the degree of the elongation depended directly 
upon the relative distance separating the two stars", and that 
the numerical values were in good agreement with Darwin's 
theoretical values for homogeneous incompressible fluids. The 
ellipticity has rarely been measured when the distance between 
the components equals or exceeds the sum of their radii. 

Russell and Shapley have also studied the distribution of 
the eclipsing binaries in space and reference to their results 
will be made in a later chapter. 

Stebbins, among others, has reminded us that every short 
period spectroscopic binary star would be an eclipsing variable 
to an observer in the plane of its orbit, and he has drawn the 
corollary, that probably a number of eclipsing variables with 
relatively small range in brightness exist among the known 
spectroscopic binaries. Acting upon this conclusion, he took 
up the examination of several such systems with the sensitive 
selenium-cell photometer which he had developed, and was 
able to show that the first two studied were in fact eclipsing 
systems. It is not to be expected that this record will be 



THE BINARY STARS I9I 

duplicated; indeed, Stebbins himself found that many of the 
systems examined later gave no evidence of light variation. 
Nevertheless, the field is a promising one, but the instruments 
demanded for its cultivation are the very sensitive modern 
photometers. Excellent work, however, may be done with any 
good photometer on the eclipsing binaries of greater light 
range, and we may expect a considerable addition to the num- 
ber of orbits of such systems in the near future if the observa- 
tions are carried out systematically. The description of the 
instruments and methods of observation may be found in 
books on variable stars like Miss Furness's "Introduction to 
the Study of Variable Stars" or Father Hagen's "Die Verander- 
liche Sterne." 

REFERENCES 

Russell, H. N. On the Determination of the Orbital Elements of 
Eclipsing Variable Stars. Astrophysical Journal, vol. 35, p. 315, 
191 1 ; vol 36, p. 54, 1912. 

Russell, H. N. and Shapley, H. On Darkening at the Limb in 
Eclipsing Variables. Astrophysical Journal, vol. 36, p. 239 and 
385. 1912. 

Shapley, H. The Orbits of Eighty-seven Eclipsing Binaries — A Sum- 
mary. Astrophysical Journal, vol. 38, p. 158, 1913. 

. A Study of Eclipsing Binaries. Contributions from the Prince- 
ton University Observatory, no. 3, 1915. 

These papers contain references to many others by the 
same, and by other writers. Attention should also be called to 
the recent Bulletins of the Laws Observatory, University of 
Missouri, for Professor R. H. Baker has entered upon an 
extensive program of work on eclipsing binaries and the results 
are being published in those Bulletins. 



CHAPTER VIII 

THE KNOWN ORBITS OF VISUAL AND 
SPECTROSCOPIC BINARY STARS 

The orbits of 112 visual and of 137 spectroscopic binary 
star systems have been computed by different astronomers,^ 
by means of the methods outHned in preceding chapters. 
Many of these orbits, especially those of spectroscopic binaries 
and of short period visual systems, may be regarded as de- 
finitive, others, especially those of the very long period 
visual binaries, have little or no value. Every computation 
was undertaken with the immediate object in view of repre- 
senting the observed motion and of predicting the future 
motion in the particular system on the assumption that the 
controlling force is the force of gravitation; but back of this 
lay the broader motive of providing additional data for the 
study of the greater questions of stellar motions and, particu- 
larly, of stellar evolution. In the present chapter I shall 
present some relations and conclusions which may be deduced, 
with more or less certainty, from the computed orbit elements, 
but their interpretation and their bearing upon the problems 
of the origin of the binary stars and their relations to single 
star systems will be left for a later chapter. 

Table I gives the elements of eighty-seven of the visual 
binary star systems, divided into two groups, the first con- 
taining the orbits which are at least fairly good approxima- 
tions, the second, the less accurate orbits. The dividing line 
is not a very definite one; several orbits included in either one 
of the two groups might with perhaps equal propriety find a 
place in the other. In each group the systems are arranged 
in the order of right ascension, and the columns, in order, give 
the star's name, the position for 1900.0, the magnitudes and 
spectral class (taken from the Harvard Photometry as far as 

1 Written in October, igi?- Tables I and II, containing the orbits of these stars, are 
placed at the end of the book for convenient reference. 



THE BINARY STARS I93 

possible), the orbit elements, and the name of the computer. 
Only one orbit is given for each system, the most recent one, 
unless there is a special reason for choosing an earlier one. 
For many systems ten or more orbits have been published. 
References to the earlier orbits may be found in Burnham's 
General Catalogue or in Lewis's volume on the stars of the Men- 
surae Micrometricae. The later orbits are, for the most part, 
published in the Astronomische Nachrichten, the Astronomical 
Journal, and Volume 12, of the Publications of the Lick Obser- 
vatory. The orbits which have been computed for twenty-five 
other systems are omitted because the data upon which they 
are based are entirely inadequate, and any conclusions drawn 
from the results would be entitled to no weight whatever. 
Perhaps the best observed among the omitted systems is the 
well known binary o- Coronae Borealis; using practically the 
same observations, Lewis finds the revolution period to be 340 
years, while Doberck's value is 1,679 years! 

For reasons already given, the orbits of the spectrograph ic 
binary stars are, in general, more accurate than those of the 
visual binaries, and only five of the 137 are assigned to the 
second group of Table IL Of these, a Persei is excluded from 
the main table because there is still a reasonable doubt as to 
whether or not it is a binary at alP; ^ Canis Majoris and 
p Leonis because the orbits are avowedly only rough approxi- 
mations, and the two long period systems, a Orionis and a 
Scarpa, because Lunt's ' recent discussion shows that the data 
on which the orbits rest are inadequate for good solutions. It 
does not follow that all of the orbits retained in the main divi- 
sion of the table are of equal value; doubtless, several should 
properly be transferred to the second section for reasons similar 
to those given for the transfer of a Orionis and a Scorpii. A 
few others depend on the H and K lines of calcium only, lines 
which exhibit anomalies not yet perfectly understood; and 
one system, <^ Persei, has recently been pronounced as still "a 
complete riddle." The results which follow, however, are not 
afifected by their retention. 

^Lick Observatory Bulletin, 7, 99, 1912. 
» Astrophysical Journal, 44, 250, 1916. 



194 THE BINARY STARS 

As in the table for the visual systems, the stars are arranged 
in order of right ascension. The columns give, consecutively, 
the star's name, the position for 1900.0, the magnitude and 
spectrum (from the Harvard Photometry) the orbit elements 
and the computer. Here again but one orbit is given, though 
two or more have been computed for several systems. In a 
number of systems, however, the second spectrum is visible 
on the plates, and values of the elements of the secondary, 
which differ from those of the primary, if given by the com- 
puter, are entered in the line below the principal orbit. In a 
few instances, also, the elements of a third body in the system, 
or of a secondary oscillation in the orbit are added. 

RELATIONS BETWEEN PERIOD AND ECCENTRICITY 

Certain striking characteristics of the orbits in the two 
tables are recognized on the most casual inspection; for ex- 
ample, the eccentricity of the visual orbits is generally large, 
that of spectroscopic orbits generally small; the periods of 
the former are long — the shortest known so far being 5.7 years 
(5 Equulei) — those of the latter generally short, ranging from 
about five hours to less than 150 days with but few exceptions. 

See, Doberck, and many other recent writers have called 
attention to the high average eccentricity of the visual binary 
star orbits, See finding in this fact strong support for his earlier 
theory (since abandoned by him, but not entirely by others) 
of the origin of the binary star systems. The average value of 
the eccentricity for the eight major planets in the solar system 
is only 0.06, and the largest, for Mercury, is only 0.206; 
whereas the average of the sixty-eight values in the first part 
of Table I is 0.48, agreeing very precisely with the value See 
found in 1896 from the orbits of forty systems. The average 
value of e for the nineteen systems in the second group is even 
higher, exceeding 0.61. 

On the other hand, the average eccentricity of the 132 orbits 
of spectroscopic binaries included in the main division of 
Table II is only 0.205, and of the five more uncertain orbits 
only 0.30. Recalling the fact that the periods of the visual 
binaries are, on the average, much greater than those of the 



THE BINARY STARS 



195 



spectroscopic, it is natural to try to establish a relation between 
the two elements. Doberck long ago presented evidence tending 
to show that the eccentricity of the visual binaries increased 
with the length of the period; Campbell, Schlesinger, Luden- 
dorff and others have shown that a similar relationship exists 
among the spectroscopic binaries. 

Examining the data now available, we find that the relation- 
ship is established beyond question. Omitting the twelve 
Cepheid variable stars ^ and RR Lyrae, a 'cluster type' vari- 
able, because the peculiarities in these systems seem to differ- 
entiate them too much from the other spectroscopic binaries 
to justify their inclusion in studies of the relations between 
orbit elements, I have classified the remaining 119 stars in the 
main part of Table II according to period and eccentricity and 
give the results in Table III. Table IV contains a similar 
grouping for the sixty-eight visual binaries of the first part of 
Table I. 

TABLE III 

Periods and Eccentricities in Spectroscopic Binaries 





d 
0-5 


d 
S-io 


d 
10-20 


d 
20-50 


d 
50-150 


d 
150+ 


Sums 


o-.io 


40 


9 


6 


3 


3 


I 


62 


.10-. 20 


5 


4 


I 





2 


4 


16 


. 20- . 30 


I 


5 


I 


I 


2 


2 


12 


.30-. 40 








2 


I 


2 


I 


6 


.40-. 50 





I 





2 


I 


3 


7 


.50-. 60 








I 


3 


3 


2 


9 


.60-. 70 








I 








I 


2 


.70-. 80 











3 


I 





4 


.80-. 90 














I 





I 


.90-1.00 























Sums 


46 


19 


12 


13 


15 


14 


119 



* /3 Cephei should perhaps be classified with the Cepheid variables, for according to 
Guthnick, the light curve resembles that of variables of this class. The variation, however, 
is only 0.05 magnitude, and in its spectral class and the characteristics of its orbit it differs 
decidedly from the twelve Cepheids which I am discussing separately. 



196 



THE BINARY STARS 



TABLE IV 
Periods and Eccentricities in Visual Binaries 



X 


y 
0-50 


y 
50-100 


y 
100-150 


y 
150+ 


Sums 


O-.IO 

















.IC>-.20 


5 


I 





I 


7 




20-. 30 


4 











4 




30-. 40 


7 


4 


2 


I 


14 




40-50 


5 


6 


I 


2 


14 




50-. 60 


4 


4 


I 


5 


14 




60-. 70 


I 


I 


I 


I 


4 




70-. 80 


3 


3 





I 


7 




80-. 90 


I 


I 





I 


3 




90-1.00 








I 





I 


Sums 


30 


20 


6 


12 


68 



A more succinct summary may be made as follows: 

TABLE V 

spectroscopic Binaries 



Number 


Average Period 


Average Eccentricity 




d 




46 


2.75 


0.047 


19 


7.80 


0.147 


12 


1517 


0.202 


13 


30.24 


0.437 


15 


106.4 


0.371 


14 


1.035- 


0.328 



Visual Binaries 



30 

20 

6 

12 




0.423 
0.514 
0.558 
0.529 



THE BINARY STARS 



197 



To smooth the relationship curve let us combine lines 3 and 
4, 5 and 6, and 9 and 10 of Table V; we then have: 



TABLE VI 





P 


e 


Sp. Bin. 46 


2.75 da. 


0.047 


19 


7.80 


0.147 


25 


23.00 


0.324 


29 


555( = i-5y.) 


0.350 


Vis. Bin. 30 


3i.3yr. 


0.423 


20 


74-4 


0.514 


18 


170 


0.539 



It is of interest, however, to note in Table V the definite maxi- 
mum of e in spectroscopic binary stars with periods of from 
twenty to fifty days, and the similar, but less marked, maxi- 
mum in visual binaries with periods between 100 and 150 years. 
The latter may be apparent, only, since but six stars are 
involved. The rapid increase of e with lengthening P in the 
first three lines of Table VI and the relatively slow increase in 
the later lines is worthy of note. 

The evidence from the less certain orbits may be added, for 
the sake of completeness ; the three short period spectroscopic 
binary stars have an average period of 5.5 days and an average 
e of 0.36; while for the two long period ones the data are 
p = 5.99 yr., 6 = 0.22. Had these results been included in 
Table V the general order of the averages would not have been 
affected. This remark holds also for the nineteen uncertain 
orbits of visual binaries, but here the last two numerical values 
would have been changed materially, for we have : 



6 stars average P 6'j .'j yr. 

6 stars average P 118 yr. 

7 stars average P 213 yr. 



average e 0.51 
average e o . 68 
average e o. 65 



The relationship so definitely established between the length 
of the revolution period and the degree of ellipticity of the 
orbit must have a physical significance. 



198 THE BINARY STARS 

RELATIONS BETWEEN PERIOD AND SPECTRAL CLASS 

Dr. Campbell, In his study of the spectroscopic binary stars 
found evidence of a relationship between the period and the 
spectral class; taking the spectra in the order B, A, F, G, K, 
and M the period increases as we pass from B toward M. 
Before analyzing the present data to see whether they support 
this conclusion, it should be said that In combining the various 
Sub-classes, I have followed the Harvard Observatory system, 
making Class B include Sub-classes O to B8, Class A, Sub- 
classes B9 to A3, Class F, Sub-classes A5 to F2, Class G, Sub- 
classes F5 to Go, Class K, Sub-classes G5 to K2, and Class M, 
Sub-classes K5 to Mb. This differs somewhat from the group- 
ing adopted by Campbell; a fact which must be kept in mind 
if comparisons are made between his tables and those which 
follow.^ It must also be noted that Table I contains a few 
exceptional stars that cannot at present be fairly considered 
in this connection. These are x Aurigae, Class Bi, whose 
period of 655 days equals more than half the sum of all the 
remaining forty-seven Class B periods; 7 Geminorum, Class 
A, period 2,175 days, and e Ursae Major is, Class A, period 4.15 
years (= 1,515 days), respectively four and three times the 
sum of the periods of the remaining thirty-six Class A stars; 
and € Hydrae, Sub-class F8, with a period of 5,588 days, as 
compared with 3,476 days for the sum of the remaining thirteen 
Class G stars. 

Omitting these four stars from Table II (e Hydrae is counted 
with the visual binaries) and, as before, the Cepheid and 
cluster-type variables, and counting the one Class M star with 
Class K, we have the following results : 

« Campbell also includes a number of systems whose periods are known either definitely 
or as 'short' or 'long', but for which no orbits have been computed. 



THE BINARY STARS 
TABLE VII 

Spectroscopic Binaries 



199 



No. 


Class 


Av. P 


' Av. e 


48 


B 


26.76 da. 


0.189 


36 


A 


13-35 


0.187 


10 


F 


32.76 


. 252 


13 


G 


267.4 


0.129 


8 


K-M 


152.9 


0.196 



Visual Binaries 



13 


A 


98.9 yr. 


0.529 


9 


F 


100.6 


0.512 


30 


G 


78.7 


0.478 


12 


K 


86.0 


0.432 


4 


M 


126.7 


0.402 



Table VII has several features of interest. In the first place, 
it appears that if we divide the spectroscopic binaries into two 
groups by a line between Classes F and G, those of the second 
group have average periods decidedly longer than the periods 
of those in the first ; but that in neither group does the period 
increase, on the average, with advancing spectral class. In 
the second place, we note that the distribution with respect 
to spectral class is very different in the two sets of binaries, 
the spectroscopic and the visual; spectra of early type pre- 
dominate among the former, whereas there is not a single Class 
B star among the latter, and nearly half the number belong to 
Class G. 

Looking over the list of spectroscopic binaries for which no 
orbits have as yet been computed, I find that in fifty- three 
cases either the approximate period or the note 'long' or 
'short' is given. Classifying these according to spectrum, I find : 

B A F G K M 



Period short 


13 


9 


6 


5 








Period long 


I 


I 





8 


7 


I 



Several years ago I also tabulated the spectral classes of 164 
visual binaries which show rapid orbital motion and found 



200 THE BINARY STARS 

that four were of Class B, 131 of Classes A to F (including 
Sub-classes F5 and F8), 28 of Classes G to K (including K5) 
and only one of Class M. 

In connection with the facts just stated it must be kept in 
mind that spectroscopic systems of short period are more 
readily discovered than those of long period, for not only is 
the amplitude of the velocity curve greater, in general, in the 
former but the variation in the velocity becomes apparent in 
a much shorter time. We may expect relatively more long 
period systems in future discoveries among stars of all classes 
of spectra and hence an increase in the average values of the 
periods. It should also be noted that the spectra of stars of 
the later types, in general, show more lines, and these more 
sharply defined, than the spectra of the early type stars. The 
probable error of measure is therefore less and hence a variable 
radial velocity of small amplitude may be unmistakable in, 
say a Class G or K star, whereas one of equal amplitude may 
escape recognition in a star of Class B. This might account 
in part, for the relative numbers of the systems of long period 
among the different classes of spectroscopic binaries, but it 
obviously does not explain the very large relative number of 
short period binaries of Class B. As for the visual binaries, 
if we accept the current doctrine that the stars of Class B 
and the brighter stars of Class M are the most remote we might 
expect that some systems of these classes with periods of the 
order of those given in the table would fall below the resolving 
power of our telescopes. Allowing for the effects of these 
factors, we may still conclude that, taking both the average 
periods and the number of systems into account and also 
the difference in these respects between the visual and 
spectroscopic systems the evidence is definitely in favor of 
an increase of the period of binary stars with advancing 
spectral class. 

In passing, attention may be called to the curious distribu- 
tion of the eccentricities in Table VII ; the definite progression, 
or rather retrogression, with advancing spectral class shown 
by the visual systems is in marked contrast with the distribu- 



THE BINARY STARS 201 

tion among the spectroscopic systems. It is doubtful whether 
any significance attaches to either. 

THE DISTRIBUTION OF THE LONGITUDES OF PERIASTRON 

In 1908 Mr. J. Millar Barr * called attention to a singular 
distribution of the values of w, the longitude of periastron, in 
those spectroscopic binaries whose orbits are elliptic. In the 
thirty orbits available to him in which e was greater than 0.0, 
twenty-six had values of co falling between 0° and 180° and 
only four between 180° and 360°. Since there is no conceivable 
relation between the position of the longitudes of periastron 
in these orbits and the position of the Sun in space (except in 
the case of the Cepheid variables), such a distribution is in- 
herently improbable unless it is produced by some error of 
observation. This, in fact, was Barr's conclusion — "the elliptic 
elements, e and w, as computed and published for the orbits 
under notice, are probably illusory, the 'observed radial ve- 
locities' upon which they are based being vitiated by some 
neglected source of systematic error." 

If this conclusion were well founded, it would be a serious 
matter indeed, but both Ludendorff and Schlesinger have 
shown that the one-sided distribution of periastra noted by 
Barr "was nothing more than a somewhat extraordinary co- 
incidence," for it becomes less marked as additional orbits are 
considered. The data in Table II tend to confirm this state- 
ment, though the inequality of distribution is not yet entirely 
eliminated. Omitting the Cepheid variables and three stars 
for which co might be taken on either side of the dividing line 
(2 Lacertae, co = 180°, /3 Lyrae, w = 0°, and w Cassiopeiae, a 
system with practically equal components either of which 
might be regarded as the primary) we have 100 systems in 
which e exceeds zero. Dividing these into two groups accord- 
ing to the value of e, the values of a? are distributed in the four 
quadrants as follows ; 

1st 2nd 3rd 4th 
c < 0.50 26 21 19 16 

c^o.50 9 3 3 3 

* Journal R. A. S. C, 2, 70, 1908. 



202 THE BINARY STARS 

The ratio of the number of oj's less than 1 80° to that of the co's 
greater than this value is thus 47 : 35 and 12 : 6 in the two 
groups respectively. Doubtless, when the number of known 
orbits is doubled, these will both be reduced approximately to 
equalities. 

THE MASSES OF THE BINARY STARS 

A knowledge of the masses of the stars is one of the funda- 
mental requirements for a solution of the mechanical problems 
of our stellar system, and this knowledge can be derived only 
from binary stars. For this reason the methods by which we 
determine the absolute and the relative masses in both visual 
and spectroscopic binaries, and the results which have been 
obtained by their application, will be presented in some detail. 

Unfortunately, the orbit elements alone do not afford all 
the data necessary for the determination of either mass or 
density. The well-known harmonic law D^ :d^= P^{M -\- Mi) : 
p'^ {m -f mi), will give the mass of any system in terms of the 
Sun's mass when the linear dimensions of the system as well as 
the orbit elements are known. But. the semi-axis major of the 
visual binary stars is known only in terms of seconds of arc, 
and its value, so expressed, must be divided by its parallax to 
reduce it to linear measure ^ ; and we do not know the true 
semi-axis major of the spectroscopic binary orbits at all, but only 
the function as'ini. This, however, is expressed in kilometers. 

The parallax of a number of visual binaries is known with a 
greater or less degree of certainty, and mass values for those 
systems may be computed, using the harmonic law in the form 

(w + Wi) = (i) 

in which tt is the parallax of the system, P the period and a 
the semi-axis major of its orbit, and the units of mass, length 
and time are respectively, the Sun's mass, the astronomical 
unit, and the year. 

While we are unable to derive the mass of any given spec- 
troscopic binary until we have a knowledge of the value of i, 

' This division gives the length in astronomical units. The astronomical unit is the 
Earth's mean distance from the Sun, in round numbers, 149,500,000 kilometers. 



THE BINARY STARS 203 

the inclination, we may nevertheless estimate the average mass 
of a number of systems with approximate accuracy, by deter- 
mining the probable average value of i and hence of sin i. 
The formulae required differ for the two cases (l) when both 
spectra have been observed, and (2) when only one spectrum 
is visible. They may be derived from the well known relation 

47r2 (a + ai)^ , X 

(m + mO = — . —jr- (^) 

in which tt denotes, not the parallax, but the circumference of 
radius unity, k the Gaussian constant (log. 8.23558), a and ai, 
the major semi-axis of the orbits of the primary and secondary 
respectively, and P their revolution period. Since we do not 
know a but only the function a sin i, we must multiply both 
members of (2) by sin^^', and since a sin i is expressed in kilo- 
meters, we must divide its value by that of the astronomical 
unit A expressed in kilometers. The numerical value of 
^.ir'^/k'^A^ is approximately 4/10^'' and we therefore have 

4 (asint + flisini)3 
(w + wi) smH = -— ' — . (3) 

Iq20 p2 

From equation (9) of Chapter IV, 

a sin i = [4. 13833] KP Vi - e 
hence 

(w + mi) sinn- = [3.01642 - 10] {K + i^i)3P(i - e^f' (4) 

the numbers in square brackets being logarithms. This equa- 
tion is independent of the parallax, or distance of the system. 
When both spectra have been measured and the correspond- 
ing velocity curves drawn we obtain at once the relative masses 
of the two components, from the relation m :mi = Ki :K\ 
and we also have the equations 

m smH = [3 . 01642 - 10] {K + K.f K,P{i - e^Y'' ) 
mi slnH = [3 . 01642 - 10] (X + KiY K P{i - e^f' ) 
from which to compute the masses of the components separately. 
When only one spectrum is visible we must apply a some- 
what different formula, namely, 

mi^ I . . . 4 (a sin iY 



sin" I = 



{m-\-mi)\ lO^o P2 



(6) 



204 THE BINARY STARS 

in which a sin i and m refer to the component whose spectrum 
is given. We may write this in a form similar to equation (4) 
thus: 

Wi' sin' i 



{m + wi)2 



[3 .01642 - loWP (i - e")*''. (7) 



In applying equations (4) and (7) it is necessary to assume 
a value for sin^i and the question of obtaining such a value has 
next to be considered. "It can be shown for an indefinitely 
great number of binary systems whose orbital planes are dis- 
tributed at random, that the average inclination would be 
57.3°, in accordance with the formula 



i smidid <i> = i, 



The average value of sin^i, however, would not be sin' 57.3° 
( = 0.65), but approximately 0.59 in accordance with the 
formula 

sin'to = — / / " sin* id id<f> = Vis tt = o . 59." 
TT J Jo 

Campbell, whom we have just quoted, and Schlesinger, who, 
from a slightly different formula obtains the same value for 
sin'i'o, point out that while this mean value holds for orbits in 
general it would not be permissible to use it for the spec- 
troscopic binary stars whose orbits have so far been computed. 
For, to quote again from Campbell, "there is the practical con- 
sideration that binary systems whose orbital planes have large 
inclinations are more readily discoverable than those whose 
inclinations are small . . . Under ordinary circumstances, 
and when dealing with a considerable number of orbits, a 
compromise value of sin^i = o . 65 might in fairness be adopted." 
For eighteen systems which he actually considers he adopts the 
higher value 0.75 because six of them are eclipsing binaries, 
with inclinations quite certainly between 60° and 90°. 

Schlesinger, assuming "that the chance of discovery is pro- 
portionate to sin i", obtains sin H = o . 68 for a mean value. 
We may then adopt, for convenience in computation, sin'^* = 
0.667 =2/3, since comparatively few eclipsing binaries are 



THE BINARY STARS 



205 



among the number under discussion. Considering first the 
spectroscopic binaries in which two spectra have been ob- 
served and for which either the values of m s'ln^i and Wisin^i 
or the ratios nti/m are given by the computer of the orbit, we 
have the data in Table VIII. 

TABLE VIII 

Relative Masses of Spectroscopic Binaries 







m sin'i 


misin'j 


m 


Wi 


m} Jm 


Boss 6142 


Bp 


18.5 


12.7 


27.8 


19.0 


0.69 


Persei 


Bi 


542 


3-79 


8.1 


5-7 


0.70 


71 Ononis 


Bi 


II. 2 


10.6 


16.8 


1.5 9 


0.95 


/3 Scorpii 


Bi 


13 


8.3 


195 


12.4 


0.64 


a Virginis 


B2 


9.6 


5-8 


14.4 


8.7 


0.60 


/3 Lyrae 


B2p 


(6.8 


16.6) 


(10.2 


24 -9) 


(2.46) 


4 Androm. 


B3 


1.50 


I. ID 


2.2 


1.6 


0.73 


X Tauri 


B3 










0.40 


u Here. 


B3 


7-5 


2.9 


(II. 2) 


(4-4) 


0.39 


57 Cygni 


B3 


1.79 


1.67 


2.7 


2.5 


0.93 


2 Lacertae 


B5 










0.81 


a Aquilae 


B8 


5-3 


4-4 


8.0 


6.6 


0.83 


V Erid. 


B9 


5-58 


5-48 


8.4 


8.2 


0.98 


yp Orionis 


A 


5-53 


4.19 


8.3 


6.3 


0.76 


136 Tauri 


A 










0.69 


/3 Aurigae 


Ap 


2.21 


2.17 


3-3 + 


3-3- 


0.98 


40 Aurigae 


A 


1-35 


I. II 


2.0 


17 


0.82 


Boss 2184 


A 


1.48 


1.27 


2.2 


1.9 


0.86 


CO Urs. Maj. 


A 










0.17 


dj Virg. 


A 










0.56 


ri Urs. Maj. 


Ap 


1.70 


1.62 


2.6 


2.4 


0.95 


e Here. 


A 


1.6 


1.0 


2.4 


1-5 


0.62 


108 Here. 


A 


0.94 


0.89 


14 


13 


0.95 


50 Drac. 


A 


0.90 


0.82 


1-4 


1.2 


0.91 


2 Sagittae 


A 


0.91 


0.65 


1-4 


1 .0 


0.72 


6 Aquilae 


A 


0.52 


0.38 


0.8 


0.6 


0.73 


b Persei 


A2 










0.28 


TT Cass. 


A5 


1.32 + 


1.33- 


2.0 


2.0 


1.003 


Boss 4423 


F 










0.96 


Leonis 


F5P 


1.30 


1 .12 


2.0 


1-7 


0.86 


d Bootis 


F5 


I 36 


1.29 


2.0 


19 


0.95 


a Aurigae 


G 


1. 19 


0.94 


1.8 


1-4 


0.79 



206 THE BINARY STARS 

Columns three and four of the table give the minimum values 
of the masses, for it is clear that m is a minimum when sin^i is 
placed equal to unity. It appears from these columns that in 
only one system, d Aquilae, may the separate components be 
regarded as probably less massive than the Sun. 

The system of /3 Lyrae is in many respects a peculiar one 
and there are exceptional difficulties in interpreting its spec- 
trum. It appears from the last column of the table that it is 
the only one in which the fainter star is definitely the more 
massive. Omitting it, the average mass ratio, rn^/in, in the 
remaining thirty-one systems is 0.748. Those who have 
examined the spectra of large numbers of stars have also 
noted, as Schlesinger says, that there appears to be a close 
correspondence between relative mass and relative brightness 
of the components; when the two spectra are almost equally 
conspicuous, the two masses are also about equal, but when 
one spectrum is barely discernible the corresponding mass is 
also relatively small. Schlesinger adds "we may infer that in 
those binaries in which the fainter component does not show 
at all, the mass of the brighter star is all the more prepon- 
derant." It must be emphasized that the numbers set down 
in the two columns m and nti have no meaning so far as any 
particular system is concerned ^ ; the value o . 667 is the mean 
value for sin'^*, but in a particular system it may have any 
value from nearly zero to unity. It is also apparent that the 
means of the two columns cannot properly be taken as the 
average masses in spectroscopic binary systems, for the indi- 
vidual results show a definite relation to the spectral class. 
Omitting j8 Lyrae, we have 





m 


Wi 


nii/m 


B to B8 (9 stars) 


12.3 


8.5 


0.69 


B9 to A5 (12 stars) 


3.0 


2.4 


0.80 


F5 to G (3 stars) 


1.9 


1.7 


0.89 



8 In the eclipsing variable u Herculis there is reason to think that i is approximately 
90°, and that the masses in columns three and four are the true masses. 



THE BINARY 


STAR 


And from the last column 






Wi/w 


B to B8 (ii stars) 


0.70 


B9 to A5 (16 stars) 


0.75 


F to G (4 stars) 


0.89 



207 



From this summary it appears that the systems of very early 
type are decidedly more massive than the others, and that all 
of the twenty-four systems for which m and mi are given are 
more massive than the Sun. There also appears to be evidence 
of an unexpected progression in the ratio of the masses of the 
two components, the secondary in systems of earlier type being 
less massive relatively to its primary, than the secondary in 
those of later type. It is somewhat remarkable that twenty- 
six of the thirty-one stars in the table have spectra of early 
type (B to A2) and that only one has a spectrum as late as 
Class G. 

These conclusions cannot legitimately be extended to all 
spectroscopic binary star systems, for the systems under dis- 
cussion are selected stars in the sense that it is only in those 
systems in which the two spectra are well separated — and the 
values of K therefore large — that the spectrum of the fainter 
component is visible. The sum {K -f K]) enters by its cube 
as a factor in equation (4), and the mass, therefore, in general, 
increases very rapidly with K. In fact, the average K for the 
primary stars in these thirty-two systems (jS Lyrae included) 
is almost precisely double that of the eighty-one single spec- 
trum systems (excluding the Cepheid variables) for which the 
computers publish this element. 

ffti siri 1 

The value of the function 7 — ; r- is frequently omitted by 

(w+wi)^ 

the computer of orbits for it gives very little definite informa- 
tion. Equation (7) affords a ready means of 'computing the 
function for any system, but I have not considered it neces- 
sary to carry out the computation, for a glance at the numbers 
recorded in Table II shows at once that no conclusions could 
be based upon any means that might be taken. For Polaris 



208 THE BINARY STARS 

this function equals 0.00,001 O ®, for 29 Canis Majoris, 4.58 
O, a range of i to 460,000; while for the exceptional system 
/3 Lyrae, it is 8.4 O. It is apparent, however, that in the 
single spectrum binaries the secondary is considerably less 
massive than its primary in nearly all cases unless we are 
willing to adopt improbably small values for (m + Wi) and 
for sin^^'. 

The present evidence may therefore be summed up in the 
general statement that in the spectroscopic binaries with 
known orbits, whether one spectrum or both spectra have 
been observed, the brighter star is, with very few exceptions, 
the more massive; binary stars of Class B have masses de- 
cidedly greater than the stars of other classes, and binary sys- 
tems of all classes have masses greater than that of the Sun. 
There is nothing novel in these conclusions; they simply con- 
firm the results obtained by earlier investigators. Ludendorff, 
for example, showed quite conclusively that among the sys- 
tems available for study in 191 1 those of Class B were, on the 
average, about three times as massive as those of Classes 
AtoK. 

Passing to the visual binaries, I have computed the masses 
of those systems for which the orbit elements and the parallaxes 
are known with sufficient accuracy to give the results signifi- 
cance. An earlier computation, in which I used every system 
for which published elements and parallaxes were available 
regardless of their probable errors, had given results which 
varied through a very wide range; but it was obvious that 
many of these were worthless either because the orbit elements 
were unreliable or because the parallax was too uncertain. 
Equation (i) shows that the parallax is the most important 
factor in this case, for it enters by its cube, while the elements 
a and P, which in a general way vary in the same sense, to a 
certain degree offset each other. Thus in the system of r? 
Cassiopeiae, an orbit with P = 345 .6 years, a = 10. 10" would 
give very nearly the same mass as the adopted orbit ; a change 
of only 0.02" or 0.03" in the value of tt, on the other hand, 
would change the mass by thirty per cent. 

» Symbol for Sun. 



THE BINARY STARS 



209 



TABLE IX 

Masses of Visual Binary Stars 



Star 



Sirius 
o2 Erid. 

8 Equulei 
Procyon 

c Hydrae 

77 Cass. 
85 Pegasi 

f Here. 

rj Cor. Bor. 

{ Urs. Maj. 

a Cent. 
70 Oph. 
Krueg, 60 
u^ Here. 



Mag. 



6, 9.0 

4.10.8 
3, 5-4 

5, 13-5 

7, 5-2 

6, 7-9 

8, II. o 

6.5 
6.1 

4 9 

1-7 

6.1 

10.8 

10.5 



Sp. 



A, A 

Ao 

F5 

FS 

F8 

F8 

G 

G 

G 

Go 
G, K5 

K 

Mb 

Mb 



49 32 
180.03 
570 
39 o 

15-3 
508. (?) 
26.3 
34 46 
41 56 
59-81 
78.83 
87.86 
54-9 
43 23 



7.55 
4-79 
0.27 
4 05 
0.23 

[2.21 
0.81 

1-35 
0.89 

2.51 
17.65 
456 
2.86 

1.30 



+0.376"' 

+0.174 

+0 . 067 

+0.31 

+0.025 

+0.201 

+0.067 

+0.107 

+0.06 

+0.179 

+0.759 

+0.168 

+0.258 

+0.106 



(m+mi) 



3.3O'" 

0.6 

2.0 

15 

3 3 

0.9 

2.6 

7 

9 



o 
6 

45 
o 



Mean mass, 14 systems 



1.76O 



Table IX gives the results of my computation, the values of 
T which were adopted being entered in the sixth column. 
With two exceptions they were derived from heliometer 
measures or the measures of photographic plates. The two 
exceptions are the parallaxes for 5 Equulei and e Hydrae. The 
former was determined by Hussey on the basis of his orbit 
and the spectrographic measures of the relative radial veloci- 
ties of the two components at the time of perihelion. The 
possibility of determining parallax by this method was pointed 
out by See more than twenty years ago, but in very few of the 
visual binary systems is it feasible to measure the relative 
velocities of the two components. The only other determina- 
tion of parallax by this method known to me is Wright's, for 
the system of a Centauri. His value, o . 73*, agrees closely with 
that obtained by the best heliometer observations. The com- 
putation is readily made by means of the following formulae, 
adapted by Wright from Lehmann-Filh^s's work " : 

" Symbol for Sun. 

" I have made slight changes in Wright's notation, which is given in Lick Observatory 
Bulletin 3, 1904. 



210 THE BINARY STARS 

Let 

R = the astronomical unit, expressed in kilometers. 
a = the semi-major axis of the binary, expressed in kilometers, 

and a'\ the same element expressed in seconds of arc. 
n = the mean angular motion of the star, in the visual orbit, in 
circular measure per second of time. 
aV = the observed difference in the radial velocity of the two 

components. 
Then 

n = 



86400 X 365 26 X P 



aFVi - c^ 



n sin i [e cos 03 -f cos(t; + ui)] 

a 



(8) 



The system of e Hydrae is unique in that the measures of the 
variable radial velocity of the bright component permit us to 
determine the orbit elements independently of the micrometric 
measures. When I computed the elements in 1912, the spec- 
trographic observations covered somewhat less than one revo- 
lution and the period was therefore assumed from the micro- 
metric measures; the elements e, T and co were found to be 
identical in the two systems; the value of i in the visual system 
permitted the separate determination of a in the spectroscopic 
system, and this, in turn, permitted the definition of a in the 
visual orbit in terms of kilometers. ^^ The result, 1,359,000,000 
kilometers, combined with the value of the astronomical unit 
in kilometers, at once gave the parallax 0.025" ^^^ hence the 
mass 3.33 O. 

f The mean value of the mass of the fourteen systems in 
Table IV is i .76 times the mass of the Sun. For the seven 
systems which he considers most reliable, Eddington ^^ obtains 
a mean value 1.66, and Innes,^^ from eight systems, a mean 

" It is to be remembered that in spectroscopic binaries with only one visible spectrum 
a is always the mean distance of the bright star from the center of gravity of the system, 
while in the visual orbit, a is the mean distance between the two components. 

IS Eddington, "Stellar Movements and the Structure of the Universe," p. 22. 

" Innes, "The Masses of the Visual Binary Stars," South African Journal of Science 
for June, 1916. 



THE BINARY STARS 



211 



value of 1 .92. I may add the testimony from the report given 
by Miss Hannah B. Steele ^^ at the nineteenth meeting of the 
American Astronomical Society, on the parallaxes of twenty 
visual binaries with known orbits measured at the Sproule 
Observatory. Of the sixteen positive parallax values, four had 
probable errors varying from one-half to four times the nu- 
merical value of the parallax. Rejecting these, Miss Steele's 
masses for the remaining twelve pairs range from 0,26 O to 
6 . 25 O with an average value of i . 7 . Admitting the meager- 
ness and the uncertainties of the data, we may still make the 
general statement that the visual binary systems for which we 
have the best orbits and parallaxes are, on the average, about 
twice as massive as the Sun. 

The danger of drawing general conclusions from this result 
arises not only from the fact that the number of stars upon 
which it is based is so small, but even more from the fact that 
they are of necessity selected stars, those relatively close to us. 
To make this clear, let us assume the mass {m + Wi) equal to 
twice the Sun's mass, and then use equation (i) to construct 
a table giving the values of a with arguments P and t, 



TABLE X 

The Semi-axis Major in a Binary Star System with Given 
Periods and Parallax 



k 


0.01' 


0.005' 


5y 


0.04—" 


. 02 — " 


10 


0.06— 


0.03 — 


20 


0.09+ 


0.05- 


40 


0.15- 


0.07+ 


60 


0.19+ 


o.io — 


120 


0.31- 


0.15+ 


250 


0.50 


0.25 


700 


0.99+ 


0.50- 


2.000 


2.00 


1.00 



Looking back to Table I, it will be found that in every single 
instance the mean distance a is greater, generally much greater, 

» Miss Steele, Popular Astronomy, February, 1917. P- 107. 



212 THE BINARY STARS 

than the mean distance in the column under ir 
Table X for the corresponding period. In other words, the 
systems listed in Table I either have more than twice the mass 
of the Sun or their parallax exceeds o.oi". Now every increase 
in our knowledge of stellar distance makes it more certain that 
the average parallax of the naked eye stars is only of the order 
of o.oi" and that the average becomes progressively smaller 
as we pass from one magnitude to the next fainter one among 
the telescopic stars, the best determinations for the stars of 
magnitudes 7, 8 and 9 being respectively about 0.009", 0.007" 
and 0.005" + - Ori the other hand, in the systems for which 
we know the parallax, the average mass is less than twice the 
mass of the Sun. We may therefore say that the visual 
binaries whose periods are known are among our nearer stellar 
neighbors. This does not hold true for the spectroscopic 
binaries because the discovery of variable radial velocity does 
not depend upon the distance of the star. 

At the Ottawa meeting of the American Astronomical So- 
ciety (191 1), Russell presented the results to which he had 
been led by employing statistical methods in a study of the 
relations between the mass, density, and surface brightness of 
visual binary stars. Hertzsprung had earlier shown the exis- 
tence of a group of stars which are entirely above the average 
in luminosity and probably in mass and had called them 
'giants' to distinguish them from the 'dwarfs'; all Class B 
stars and some stars in every one of the other spectral classes 
are placed among the giants. Dividing 160 giant binary stars 
into four groups according to spectrum, Russell determined 
the mean mass of a system in each group, the values ranging 
from 7 to 13 times that of the Sun, while the mean luminosities 
ranged from 130 to 195 times that of the Sun, an indication 
that these stars have great volumes and correspondingly low 
densities. Similarly, for the mean masses of the systems in 
five groups of 'dwarf binaries (189 systems in all), the values 
were found to range from 5.4 O for the Class A group to o . 4 
O for the Class M group, the mean luminosity decreasing in 
the same direction from 25 to 0.02 times that of the Sun, 
Making every allowance for the uncertainties introduced by 



THE BINARY STARS 213 

errors in the data and by estimating the average ^values of 
unknown functions on principles of probability, the results of 
such determinations may still be regarded as corroborative 
evidence which increase our confidence that the masses found 
by direct methods, in the few systems where such methods are 
applicable, are fairly representative of those in binary systems 
in general. 

Hertzsprung has shown that we may use equation (i) to 
determine a minimum value for the parallax of visual double 
star systems in which orbital motion has been observed but for 
which the observed arc is too short to permit the computation 
of orbit elements. 

Let 

be the orbital velocity in a circular orbit ; then from equation 
(i) we have 

where p is the parallax of the star. Now the velocity F in a 
parabolic orbit equals Fi\/2, hence, using R for the radius 
vector instead of a, we have 

But in an elliptic orbit, such as we assume for a double star, 
the orbital velocity must be less than the parabolic velocity, 
and therefore 



P'> 



8ir2(m4-wi) 



and since projection can only shorten the radius vector and 
diminish the apparent orbital velocity we must have, a fortiori, 

pz > ^J^ (9) 

87r2(w + mi) 

where r and v are the projected values of R and V. The right 
hand member of (9) is therefore the expression for the minimum 



214 THE BINARY STARS 

possible parallax, and when an assumption is made as to the 

mass, all terms in it are known, for r is given by the observed 

dd 
distance p and v^ by the observed angular velocity -— • Hertz- 

dl 

sprung assumes {m + mi) = the Sun's mass = i and writes 
for the minimum hypothetical parallax 

P\ min =— • (lO) 

oir 

Comparison with stars whose parallaxes are known leads him 
to conclude that the ratio p : ph, mm does not vary greatly 
and that, in the mean 

P 
log — =+o.27±o.i4 (ii) 

ph, min 

or, in words, the true parallax of a double star system is ap- 
proximately double the minimum hypothetical parallax, the 
probable error being about one-third of its value. From (9) 
and (10) we obtain 



as the expression for the minimum mass in a system of known 
parallax. If we accept the relation expressed by (11) it follows 
that the mass of an average double star system exceeds one- 
eighth the mass of the Sun. This may be regarded as another 
bit of evidence favoring the conclusion that the stars in binary 
system are of the same order of mass as the Sun. 

The orbit elements of a visual binary give us no direct infor- 
mation as to the position of the center of gravity of the system, 
nor as to the relative masses of the two components; but 
under favorable conditions this information may be acquired 
from measures connecting one of the components with one or 
more independent stars. When such measures, covering a 
sufficient time interval, are available for a system in which the 
angular separation is fairly large the relative masses can be 
determined in a very simple manner.^® 

i*See Astrophysical Journal, 32, 363, 1910. 



THE BINARY STARS 215 

Let AB be the binary system, C an independent star, and 
let p, 6 and p', d', respectively, be the distance and position 
angle of C referred to A and of B referred to A. Then the 
apparent rectangular coordinates of C and B referred to axes 
drawn from A as origin in position angles 60 and (90° + ^o) 
will be 

X = p cos(d — do) x' = p'cos{d' — do) 

y = p sin(e — ^o) y' = p's\n{d — Bo) 

Now if we let K equal the mass ratio t — r , the coordinates 
^ A + B 

of the center of gravity of AB will be Kx\ Ky', and since the 

motion of C with respect to this point must be uniform we have 

x = a + b{t- to) + Kx'; y = a' + h'{t - Q + Ky' (12) 

to being any convenient epoch. 

Each set of simultaneous observations of AB and AC fur- 
nishes an equation of condition in x and one in y for the deter- 
mination of the five constants a, 6, a' , h' , K. No knowledge 
of the period or other elements of the binary system is involved, 
the accuracy of the determination of K depending entirely 
upon the amount of departure from uniformity of motion of B 
relatively to A. In Lick Observatory Bulletin No. 208 I have 
published a list of systems specially suited to the application 
of this method and have urged the desirability of measuring 
them systematically. 

Van Biesbroeck ^^ has recently proposed a method equally 
simple by which the mass ratio in visual binary systems may 
be determined from measures of photographs taken with long- 
focus telescopes, and has added a list of stars to which it may 
be applied with prospects of good results within comparatively 
few years. Up to the present time, however, our information 
of the relative masses in visual systems has been derived 
almost entirely from meridian circle observations of the abso- 
lute positions of one component or of both components com- 
bined with the orbit elements. 

The most reliable values are those deduced by the late Lewis 
Boss and published in his Preliminary General Catalogue of 

'^''Astronomical Journal, 2Q, 173, 1916. 



2l6 



THE BINARY STARS 



Stars for igoo . o. Adding a few others that are fairly reHable, 
we have the data in Table XL 



TABLE XI 



star 


m' fm 


Computer 


17 Cassiopeiae 


0.76 


Boss 


Sirius 


0.29 


Boss 


Procyon 


0.33 


Boss 


f Cancri 


I. 


Seeliger 


e Hydrae AB 


0.9 


Seeliger 


^ Urs. Maj. 


I .0 


Boss 


7 Virginis 


I.O 


Boss 


a Centauri 


0.85 


Boss 


^ Bootis 


0.87 


Boss 


f Herculis 


043 


Boss 


70 Ophiuchi 


0.82 


Boss 


Krueger 60 


0.56 

o.35\ 


Russell, 2 solutions 


85 Pegasi 


1.0 


Boss 



The testimony of this table is in harmony with that afforded 
by the spectroscopic binary stars, namely, that the brighter 
star of the system is generally the more massive ; but it is only 
fair to add that other computers, notably the Greenwich ob- 
servers, obtain results that for some of these stars differ widely 
from Boss's; also that Boss himself, in the case of 85 Pegasi, 
obtained a value 1.8, but considered the uncertainties to be 
so great that he was justified in adopting i .0. 



DENSITIES OF THE BINARY STARS 

The densities of the stars in eclipsing binary systems of 
known orbits may be computed, as has been shown in the 
preceding chapter, if the ratio of the masses of the two com- 
ponents is also known, and Shapley's |extensive investiga- 
tions indicate that the average density is small. Assuming 
the disks darkened to zero at the edge, he finds the Class A 
stars to be about i /14, the Class B stars i /ii, and the Class 



THE BINARY STARS 217 

F, I /3 as dense as the Sun ; while the few stars of Class G 
for which orbits are known exhibit so great a range in density 
that average values would have no meaning. Shapley has 
given for the upper limit of the mean density in an eclipsing 
system, the simple formula ^^ 



do< 



0.054 



P2 sin3 ^' 



in which P, the period and t, the semi-duration of the eclipse, 
are expressed in days and do in terms of the Sun's density. 
Applying it to five systems of spectral Class F8 to G5, he 
obtains upper limits ranging from 0.02 O to 0.00005 O. It 
is to be remembered that Shapley, throughout, assumes the 
equality of the masses of the two components. 

We cannot proceed so simply in the case of the visual bi- 
naries and those spectroscopic binaries which are not also 
eclipsing variables because the orbit data do not include any 
information as to the diameter of the disks of the component 
stars. When, however, in addition to the orbit elements, a 
and P, we know the surface brightness (which is a function of 
the absolute temperature and thus of the spectral class), and 
the ratio of the masses of the components, the density may 
be computed. Such a computation has recently been carried 
out by E. Opik^^ for the principal component (both components 
in the system a Centauri) of each of thirty-nine visual binaries. 
The results obtained are necessarily only rough approxima- 
tions because all of the data are more or less seriously affected 
by errors of observation. Nevertheless, they are of decided 
interest, for it is probable that in their densities as in other 
physical conditions, the visual binaries are more nearly rep- 
resentative of the average stars (excluding the 'giant* stars) 
than the eclipsing binary systems. The extreme range in 
Opik's tables is from 0.012 to 5.9 the Sun's density, twenty- 
six values fall between o. 16 and i .45, and the average of the 
forty is 0.39. A relation between spectral class and density 
is indicated, the stars of Class Ao — A5 being the densest 

^* Astrophysical Journal 42, 271, 191S. 
1* Astrophysical Journal 44, 292, 1916. 



2l8 THE BINARY STARS 

(o . 65 O ) and those of Class K — K5 the least dense (o . 072 O ) ; 
Class B is not represented. 

I have already mentioned Russell's statistical studies which 
lead him to assign extremely small densities and high lumin- 
osities to some of the giant stars and high densities and low- 
luminosities to some of the dwarfs. Weighing the evidence 
given by the visual as well as by the spectroscopic binaries, 
it appears that while we may regard our Sun as a fairly typical 
star in point of mass, it is hardly possible to use the expression 
'typical star' when we speak of density or luminosity. Con- 
sider Sinus, for example; the bright star is only three and one- 
half times as massive as its companion and about two and 
one-half times as massive as the Sun, but it is more than 1 1,000 
times as luminous as the former and fully thirty times as 
luminous as the latter. 

THE PARALLELISM OF THE ORBIT PLANES OF THE 
VISUAL BINARY STARS 

A number of investigations have been made to ascertain 
whether the orbit planes of the visual binary stars exhibit a 
random distribution or whether there is a tendency to par- 
allelism to a particular plane, as for example, the plane of the 
Milky Way. These investigations have, as a rule, been based 
upon the systems whose orbits are known although the fact 
that the orbit elements do not define the plane uniquely 
presents a serious difficulty. In only three or four cases has 
the indetermination in the sign of i, the inclination, been 
removed by spectrographic measures, and the true pole of the 
orbit thus distinguished from the 'spurious' pole. Miss Everett,^^ 
See 2^ and Doberck^^ reached negative conclusions; their 
researches gave no definite evidence that the poles of the orbits 
favored any special region of the celestial sphere. Lewis and 
Turner 2^ concluded that the evidence indicated, though some- 
what doubtfully, a tendency of the poles to group themselves 

"Alice Everett, Monthly Notices. R. A. S., 56, 462, 1896. 
" T. J. J. See. Evolution of the Stellar Systems, i, 247, 1896. 

«W. Doberck, Astronomische Nachrichten, 147, 251, 1898, and Astronomische Nach- 
richten, 179, 199. 1908. 

» T. Lewis and H. H, Turner, Monthly Notices, 67, 498, 1907. 



THE BINARY STARS 219 

on or near the Milky Way. Bohlin^^, on the contrary, reached 
the conclusion that the poles may be divided into two groups, 
one favoring a point near the pole of the galaxy, the other a 
point near the pole of the ecliptic and the apex of the Sun's 
way. Professor J. M. Poor^^ has recently attacked the problem 
by a different method based upon the thesis that "were the 
orbit-planes of binary stars parallel, then because the apparent 
orbits of those situated on the great circle parallel to their 
orbit-planes would be straight lines, while at the poles of this 
great circle the apparent orbits would be ellipses, the parallel- 
ism would show itself in a statistical study as a variation in 
correlation between position angle and distance of doubles in 
different parts of the sky." Using the data given in Burnham's 
General Catalogue and the later lists of double stars (to 1913) he 
reached the conclusion that a 'preferential pole' near the ver- 
tex of preferential motions of the stars was indicated. 

In view of these divergent results we may consider the ques- 
tion as one of the many which still remain to be answered. 

THE CEPHEID VARIABLES ^^ 

The Cepheid variables entered in Table II have been omitted 
from the later tables because, considered as binary systems, 
they seem to belong in a class by themselves. Every variable 
of this type which has been investigated with the spectrograph 
has shown a variable radial velocity and the period indicated 
by the velocity curve has in every instance been equal to the 
period of light variation. On the one hand this has been 
regarded as sufficient proof that these variables are binary 
systems; on the other, that the light variation is in some 
manner caused by the interaction of the two components. 
But many difficulties are met in attempts to construct a theory 
for the nature and cause of the variation, and no theory that 
is satisfactory in all respects has as yet been formulated. 
Eclipse phenomena certainly do not enter, at least in the pro- 

** K. Bohlin, Astronomische Nachrichten, 176, 197. 1907. 
" J. M. Poor, Astronomical Journal, 28, 14s, 1914- 

»• I include under this head the Geminid variables also, for the two classes have no real 
dividing line. 



220 THE BINARY STARS 

duction of the principal minimum, for the epoch of this mini- 
mum does not coincide even approximately with an epoch of 
zero relative radial velocities of the components. On the 
contrary, it has been shown by Albrecht and subsequent inves- 
tigators of orbits of these stars that the epoch of maximum 
light agrees closely with the time of maximum velocity of 
approach of the bright star, and the epoch of minimum light 
nearly as closely with the time of maximum velocity of reces- 
sion; that is, as Campbell puts it, the epochs of maximum 
and minimum light are functions of the observer's position in 
space. Certain observed irregularities in the light curves of 
many of the Cepheids and in the velocity curves of several 
for which orbits have been computed, the changes in color 
and spectral type which the Mount Wilson observers have 
shown to accompany the light variation, and the independent 
determinations by Hertzsprung and Russell that these stars 
are of great absolute brightness and probably of very great 
volume are additional elements of difficulty. In fact, some 
astronomers have raised the question whether the observed 
line-displacements in the spectra of these stars really indicate 
orbital motion in a binary system or whether they may not 
have their origin in physical conditions prevailing in the atmos- 
pheres of single stars. The majority of astronomers, however, 
still hold to the opinion that they are binary systems. 

Adopting the latter view, let us examine the characteristics 
of the computed orbits of the twelve Cepheid variables in 
Table II and of RR Lyrae, a 'cluster- type' variable. This star 
is properly included with the Cepheids for it is becoming 
apparent that the chief distinction between the two classes 
arises from the very short periods of the cluster variables. ^^ 
The star /3 Cephei is probably a Cepheid variable, but, as 
already noted, it differs entirely in its spectrum and in most 
of the characteristics of its orbit from the other systems of 
this class, and closely resembles in these particulars the short 

" RR Lyrae, however, has an annual proper motion of o.2s'±, which is so much 
greater than that of the average Cepheid that, on the usual assumptions as to the relations 
between parallax and proper motion, its luminosity is of the order of the Sun's, while the 
Cepheids, according to Hertzsprung, exceed the Sun in absolute brightness by about seven 
magnitudes. 



THE BINARY STARS 



221 



period binaries which show no light variation. I have there- 
fore included it with the latter and omit it here. 

TABLE XII 

Cepheid Variable Stars in order of Revolution Period 



Star 


Sp. 


P 


e 


(a 


RR Lyrae 


F 


o.567d 


0.271 


96.85° 


SZ Tauri 


F8 


3 


148 


0.24 


76.66 


RT Aurigae 


G 


3 


728 ■ 


0.368 


92.016 


SU Cygni 


F5 


3 


844 


0.21 


(345-8) 


Polaris 


F8 


3 


968 


0.13 


80.0 


T Vulpec 


F 


4 


436 


0.440 


104.03 


5 Cephei 


G 


5 


366 


0.484 


85-385 


Y Sagit. 


G 


5 


773 


0.16 


32.0 


X Sagit. 


F8 


7 


012 


0.40 


93-65 


1) Aquilae 


G 


7 


176 


0.489 


68.91 


W Sagit. 


F5 


7 


595 


0.320 


70.0 


f Gemin. 


G 


lO 


154 


0.22 


333- 


YOph. 


G 


17 


121 


0.163 


201.7 



166,000km 

460,000 

856,000 

1,350,000 

164,000 
966,000 

1,270,000 

1,485,000 

1,334,000 
1,773,000 

1,930,000 
1,798,000 
1,790,000 



Wi^sin't 
(m-fmi)' 



O . 0006 

o . 0004 

0.0018 

0.0058 

o.ooooi 

0.0018 

o . 0030 

o . 0040 

0.0016 

0.0043 

0.0050 

0.0023 

o . 0008 



The spectra of all thirteen stars fall within the limits F to G 
of the Harvard scale. In general, variables of this type are 
almost wholly unknown among stars of spectral Classes B, 
A, M or N. Tabulating the data for the fifty-three known in 
1910, Campbell found one of Class A, forty of Classes F to K5, 
and twelve of unknown class. 

From the column e in the table it appears that the relations 
between eccentricity and period which have been established 
for the other binaries do not hold good for these systems, the 
average eccentricity of the thirteen (average period 6. 15 days) 
being o . 300, more than double that given in Table V for the 
systems with average period 7 . 8 days. Nor is the increase of 
eccentricity with period very definite, though if we divide the 
stars into three groups according to period (o to 5, 5 to 10, 
and 10 to 20 days) we find: 



6 stars Av. P = 
5 stars Av. P = 
2 stars Av. P = 



3 . 28 days 

6 . 58 days 

13.64 days 



Av. c = 0.276 
Av. e = 0.371 
Av. e = 0.192 



222 THE BINARY STARS 

The relation between period and spectral class is necessarily 
quite indeterminate; the last seven stars in the table, however, 
on the average, belong to a somewhat later spectral class than 
the first six. 

The values of a sin i are all less than 2,000,000 kilometers, 
"which is evidence," in Campbell's words, "that the primaries 
revolve in orbits whose dimensions may be described as 

ffii Sin T- 

minute." All of the values of ; — are also- remarkably 

(m + wi)2 

small, the largest being less than 0.006 O and the average for 
the thirteen only 0.0024 O, which is far below the average 
for the other spectroscopic binaries. Three factors enter into 
this quantity, the orbital inclination, the mass of the system, 
and the ratio of the masses of the components. That the 
resulting function should in every case be so small argues 
again for the similarity of physical conditions in all of these 
systems. If the light variation is due either directly or indi- 
rectly to the binary character of these systems it is highly 
improbable that the inclinations of the orbit planes are small, 
but if we adopt for each of these systems the value sm^i = 0.667 

1 . f t . rr^ 1 1 -.TTTT 1 r • wi' siii^^* 

which was used m Table VIII, the function- — ; r^ior the 

thirteen has an average value of only 0.0036 O and we must 
conclude either that the systems are much less massive than 
the Sun or that the ratio of the mass of the secondary to that 
of the bright star is very small. Ludendorff, among others, 
has shown that the former alternative is entirely improbable; 
but if the bright star is to equal the Sun in mass, it must, to 
produce the average value just given, be six times as massive 
as its secondary, a ratio far greater than that established in 
Table VIII. 

Finally, we may note that in ten of the systems the value 
of CO falls not only in the first or second quadrant, but between 
the narrower limits 32° and 104°, the average being 80°. 
According to Curtiss this grouping, first pointed out by Luden- 
dorff", is to be expected; for the observed tendency to syn- 
chronism between the epochs of light maximum and of maxi- 
mum velocity of approach, and between the epochs of light 



THE BINARY STARS 223 

minimum and of maximum velocity of recession, combined 
with the well-known tendency toward rapid increase and slow 
decrease of light in variables of this type must create a ten- 
dency toward the location of periastron on the descending 
branch of the velocity curve. Curtiss questions the correct- 
ness of the published value of co for SU Cygni and gives plaus- 
ible reasons for the discrepant values of this element in the 
last two systems of the table. 

MULTIPLE STARS 

In 1 78 1, Herschel noted that the brighter star of the 5" pair, 
f Cancri, discovered by Tobias Mayer in 1756, was itself a 
double star with an angular distance of only i" between its 
nearly equal components. In the years that have followed, a 
large number of such triple systems, and not a few that are 
quadruple, or multiple, have become known. During the Lick 
Observatory double star survey, for example, I catalogued at 
least 150 such systems previously unknown, and Professor 
Hussey's work yielded a proportionate number. The triple 
was formed, in more than half of these cases, by the discovery 
of a close companion to one of the components of a wider pair 
previously catalogued by other observers, and in some cases 
there is no question but that the closer pair had been over- 
looked at the earlier date because it was below the resolving 
power of the telescope. 

The spectrograph has also revealed many triple and mul- 
tiple systems; sometimes, as in 13 Ceti or k Pegasi, by show- 
ing that one component of a visual binary is itself a binary 
too close to be seen as such with the telescope; again, as 
in Polaris, by showing that the short period spectroscopic 
binary revolves in a larger orbit with a third invisible star. 
Though I have made no complete count, I think it a fair esti- 
mate that at least four or five per cent, of the visual binaries 
are triple or quadruple systems. It seems to be a general rule 
that the distance between the components of the close pair 
in such systems whether visual or spectroscopic is small in 
comparison to that which separates the pair from the third 
star, and an argument has been based upon this fact to support 



224 THE BINARY STARS 

a particular theory of the origin of binary systems, as we shall 
see in a later chapter. However, there are exceptions to the 
rule. Thus we have in Hu 66, BC = 0,34", A and BC ( = 
02351) =0.65"; in A 1079, AB = o.23^ABandC = 0.48"; 
in A 2286, AB = 0.34", AB and C = 0.94"; in A 1813, 
AB = 0.20'', AB and C = 0.70"; and in HU91, BC = 0.15", 
AB ( = 02 476) = 0.54'. The system of Castor affords an 
extreme example of the contrasting distances between the 
close and wide pairs in a quadruple star; each component of 
the visual pair is a spectroscopic binary, the revolution periods 
being respectively three and nine days while the period of the 
orbit described by these two pairs is certainly greater than 
300 years! The motion of the third star with respect to the 
closer pair in a triple visual system has in no instance been 
observed over an arc long enough to permit the computation 
of a reliable orbit. 

The results for the mass and density of the binary stars, and 
for the relations between the orbital elements which have been 
set forth on the preceding pages rest upon comparatively 
small numbers of stars, and these are, to a certain degree at 
least, selected stars, as has been remarked. When the number 
of reliable orbits has been doubled, as, from present indications, 
it will be within two or three decades, some of them may 
require modification; many of them, however, may be ac- 
cepted as already definitely established not only for the systems 
upon which they are based, but for binar>^ systems in general. 

REFERENCES 

In addition to the papers cited in the footnotes to the chap- 
ter, the following will be of interest: 

Campbell, W. W. Second Catalogue of Spectroscopic Binar>' Stars, 
Lick Observatory Bulletin^ vol. 6, p. 17, 1910. 

ScHLESiNGER, F., and Baker, H. A Comparative Study of Spectro- 
scopic Binaries. Publications of the Allegheny Observatory, vol. J, 
p. 135, 1910. 

LuDENDORFF, H. Zur Statistik der Spektroskopischen Doppelsterne, 
Astronomische Nachrichten, vol. 184, p. 373, 1910. 



THE BINARY STARS 225 

LuDENDORFF, H, tlber die Massen der Spektroskopischen Doppelsterne, 
Astronomische Nachrichten, vol. i8g, p. 145, 191 1. 

Hertzsprung, E. Uber Doppelsterne mit eben merklicher Bahn- 
bewegung, Astronomische Nachrichten, vol. igo, p. 113, 1912. 

AiTKEN, R. G. A Catalogue of the Orbits of Visual Binary Stars, Lick 
Observatory Bulletin, vol. 2, p. 169, 1905. 

. Note on the Masses of Visual Binary Stars, Popular Astron- 



omy, vol. 18, p. 483, 1910. 

Russell, H. N. Relations between the Spectra and Other Charac- 
teristics of the Stars, Popular Astronomy, vol. 22, Nos. 5, 6, 1914. 

Laplau-Janssen, C. Die Bewegung der Doppelsterne, Astronomische 
Nachrichten, vol. 202, p. 57, 19 16. 

Loud, F. H. A Suggestion toward the Explanation of short-Period 
Variability, Astrophysical Journal, vol. 26, p. 369, 1907. 

Duncan, J. C. The Orbits of the Cepheid Variables Y Sagittarii and 
RT Aurigae; with a Discussion of the Possible Causes of this Type 
of Stellar Variation, Lick Observatory Bulletin, vol. 5, p. 91, 1909. 

Roberts, A. W. On the Variation of S Arae, Astrophysical Journal, 
Yo\. 33, p. 197, 1911. 

Shapley, H. On the Nature and Cause of Cepheid Variation, Astro- 
physical Journal, vol. 40, p. 448, 1914. (Contains very full refer- 
ences to other papers on this, and allied subjects.) 

CuRTiss, R. H. Possible Characteristics of Cepheid Variables, Publi- 
cations, Astronomical Observatory, University of Michigan, vol. i, 
p. 104, 1912. 



CHAPTER IX 

SOME BINARY SYSTEMS OF SPECIAL INTEREST 

Having studied the orbit elements of the binary stars in 
their more general relations, it will be of interest next to con- 
sider the various systems in themselves, the extent, and the 
limitations, of our knowledge of their motions and physical 
conditions. Selection is here an obvious necessity, and in 
making my choice I have been influenced in part by the his- 
torical associations connected with certain systems, in part 
by the peculiarities of the orbit. Some of the systems are 
among those for which our knowledge is relatively full and 
exact; others present anomalies still more or less bafifling to 
the investigator. 

a CENTAURI 

Our nearest known stellar neighbor, a Centauri, is a system 
of more than ordinary interest. One of the first half dozen 
double stars to be discovered — the very first among the stars 
of the southern heavens — it also divides with 6i Cygni the 
honor of being the first whose approximate distance, or par- 
allax, became known. It consists of two very bright stars, 
0.3 and 1.7 magnitude, respectively, which revolve in a 
strongly elliptic orbit so highly inclined to the plane of pro- 
jection that at times they are separated by fully 22", at others 
by less than 2 ". 

Accurate micrometer measures of relative position begin 
only with Sir John Herschel, in 1834, but meridian circle ob- 
servations date back to Lacaille's time, 1752. Since these early 
dates the system has been observed regularly with meridian 
circle, micrometer, and heliometer, and the position of its com- 
ponents has been measured on photographic plates. The 
material is therefore ample for a very good determination of 
the orbit elements and of the proper motion of each com- 



THE BINARY STARS 227 

ponent and excellent use has been made of it. The successive 
sets of elements by Roberts, See, Doberck and Lohse are in sub- 
stantial accord and the orbit is probably as well known as that 
of any visual binary star. The parallax is known with equal 
precision; the value resulting from the excellent heliometer 
measures by Gill and Elkin would alone assure that ; but this 
value has been confirn;ed by the accordant results by Roberts 
from the discussion of meridian circle observations and those 
by Wright from measures of the relative radial velocities of 
the components, to which reference has been made on an earlier 
page. The spectrograph has also given us the radial velocity 
of the center of mass of the system, —22.2 kilometers per 
second (in 1904). 

Taking the orbit data in Table I of the preceding chapter, 
and the parallax, 0.76", we find that the major semi-axis of the 
system equals 23.2 astronomical units; but since the eccen- 
tricity is 0.51, at periastron the components are separated by 
only 11.4, at apastron by fully thirty-five astronomical units; 
at the former time, that is, they are nearly as close together 
as Saturn and the Sun, and at the latter, farther apart than 
Neptune and the Sun. 

Wright's measures of radial velocity were made in 1904 
when the two components were near the nodal points. They 
showed that at that time the fainter component was ap- 
proaching, the brighter one receding from the Sun, relatively 
to the motion of the center of mass of the system. The former 
was therefore at the descending, the latter at the ascending 
node in the relative orbit; and hence, on the system of nota- 
tion adopted in this book, the angle 25°, given for ft in the 
table of orbits, is the ascending node and the algebraic sign of 
the inclination is positive, a Centauri is therefore one of the 
very few visual binary systems for which the position of the 
orbit plane has been uniquely determined. 

According to the adopted values for the parallax and orbit 
elements, the total mass of a Centauri is almost precisely twice 
that of the Sun, and all investigators of the proper motions of 
the two components agree that the brighter is very slightly 
the more massive of the two. Since the spectrum of this com- 



228 THE BINARY STARS 

ponent is also practically identical with the solar spectrum, 
it is frequently referred to as a replica of the Sun. Further, 
if we assume o.o as the Sun's absolute magnitude (that is, its 
apparent magnitude were it removed to a distance correspond- 
ing to a parallax of I'O, at the distance of a Centauri it would 
shine as a star of +0.6 magnitude or somewhat less brightly 
than a\ Centauri. Hence, if the luminosity per unit of surface 
area is the same in the two cases, as the similarity of the spectra 
would lead us to expect, a\ Centauri must be rather larger and 
less dense than the Sun. The spectrum of ag Centauri is of 
later type, and this fact as well as its magnitude, 1.7, indicates 
that its luminosity is less than that of its primary; probably 
it is at least as dense as the Sun. 

Finally, the accurate value of the proper motion of the 
system, 3.688" (corresponding to a velocity of about twenty- 
three kilometers per second or a little less than five astronom- 
ical units per year), combined with the value of the radial 
velocity given above shows that the system is rushing through 
space with a velocity of thirty-two kilometers per second, 
which carries it about seven times the distance from the Earth 
to the Sun in a year. This is fully sixty per cent, greater than 
the motion of translation of our solar system. 

SIRIUS 

Several references have been made to Sirius on the earlier 
pages of this volume but it will not be amiss to give a more 
Connected account of the star here. It was in 1834 that Bessel 
noticed that the proper motion of Sirius, the brightest star in 
the sky, was variable. Six years later he noted a similar 
phenomenon in the proper motion of Procyon, and by 1844 he 
had worked out the nature of the variation sufficiently to 
become convinced that it was due in each instance to the 
attraction of an invisible companion. His famous letter to 
Humboldt on the subject has often been quoted : "I adhere", 
he wrote, "to the conviction that Procyon and Sirius are 
genuine binary systems, each consisting of a visible and an 
invisible star. We have no reason to suppose that luminosity 
is a necessary property of cosmical bodies. The visibility of 



THE BINARY STARS 229 

countless stars is no argument against the invisibility of count- 
less others." 

Peters examined the existing meridian circle observations in 
1 85 1 and concluded that they supported Bessel's hypothesis; 
ten years later, T. H. Safford repeated the investigation and 
"assigned to the companion a position angle of 83.8° for the 
epoch 1 862. 1." The most complete discussion, however, was 
that of Auwers, who "placed the question beyond doubt by 
determining the orbits and relative masses of the bright star 
and the invisible companion ; but before the results were pub- 
lished, Mr. Alvan G. Clark discovered the companion, in 1862, 
near its predicted place." Bond's measures for the epoch 
1862.19, in fact, placed the companion 10.07" from the primary 
in tjie position angle 84.6°. 

Since that time it has described more than an entire revo- 
lution and the orbit elements, now known with high precision, 
agree as well as could reasonably be expected with Auwers's, 
computed before the companion's discovery. The revolution 
period, for instance, is 49.32 years according to Lohse; Auwers's 
value was 49.42 years. The eccentricity of the true orbit is 
greater than that for the orbit of a Centauri, but the inclina- 
tion of the orbit plane is considerably less and the apparent 
ellipse is therefore a more open one, the maximum apparent 
separation of the components being about 1 1. 2" and the mini- 
mum a little less than 2". The bright star is so exceedingly 
brilliant, however, that it is impossible to see the faint com- 
panion with any telescope when it is near its minimum dis- 
tance. Thus, periastron passage occurred early in 1894, but 
the last preceding measure was Burnham's in the spring of 
1892 when the angular separation was 4.19", and the little star 
was not again seen until October, 1896, when my first measure 
gave an apparent distance of 3.81". 

The magnitude of Siritis, on the Harvard scale, is —1.58; 
estimates of the brightness of the companion vary, but it is 
probably not far from 8.5 on the same scale, a difference of 
10. 1 magnitudes. Accepting these figures. Sinus radiates more 
than 11,000 times as much light as its companion; and, if the 
parallax 0.376" is correct, fully thirty times as much as our 



230 THE BINARY STARS 

Sun. Yet, according to the best mass determinations, the 
bright star is only 2.56, the companion 0.74 times as massive 
as the Sun; and Adams finds that the small star, which to the 
eye seems decidedly the yellower, has the same spectrum 
(Class A) as the bright star. These are facts which, as Camp- 
bell says, "we are powerless to explain at present." 

KRUEGER 60 

The system known as Krueger 60 (the closer pair was really 
discovered by Burnham in his careful examination of all the 
double stars noted by Krueger in the course of his meridian 
circle observations) offers a strong contrast to the two we 
have been considering, not only in its appearance but in many 
of its physical characteristics, though like them it is remark- 
able for its large proper motion and its large parallax. Sirius 
and a Centauri are two of the brightest stars in the sky and 
are also of great absolute brilliance; Krueger 60 is only of the 
ninth magnitude, despite its large parallax, and is among the 
feeblest of known stars in its actual radiating power. The 
orbit elements of the former two are known with accuracy; 
the companion star in Krueger 60 has been observed through 
less than one-third of a revolution, and there is, moreover, a 
most unfortunate gap in the series of measures from 1890 to 
1898. It is therefore surprising that we have any orbit for it 
at all. The star was neglected at first because the average 
pair of ninth magnitude stars with a separation of 2.32", such 
as Burnham's measure in 1890 gave for Krueger 60, does not 
change perceptibly in a century, and the exceptional character 
of this pair was not recognized until Doolittle measured it 
again in 1898. Since that time Barnard and Doolittle have 
measured it systematically. An excellent idea of the telescopic 
appearance of the system and of the rapidity of the orbital 
motion is given by the photographs taken in 1908 and 1 91 5 
by Professor Barnard, who has kindly permitted me to reprint 
them here. The photographs also show a third star which is 
independent of the binary and is being left behind by the latter 
in its motion through space. 



THE BINARY STARS 231 

Though the orbit elements are necessarily rather uncertain, 
Russell has been able greatly to limit the range of possible 
solutions by a skilful use of the dynamical relations connecting 
the observed coordinates (position angle and distance) and 
the times of observation. Thanks to this and to the series of 
measures connecting the star A of the binary with the inde- 
pendent star, and to our very precise knowledge of the value 
of the parallax, which three unusually accordant determina- 
tions by Barnard, Schlesinger and Russell fix at +0.256", our 
acquaintance with the physical conditions in the system is far 
more complete than so short an observed arc would ordinarily 
make possible. The period given in my table of orbits may 
indeed be in error by as much as eight years, the eccentricity 
by one-third or more of its whole amount; nevertheless, the 
mass of the system is very well determined, the value, 0.45 
being correct probably to within ten per cent. The mass ratio 
is somewhat more uncertain and may lie anywhere between 
0.36 and 0.56; that is, the mass of the brighter star is from 
3/10 to I /3, the mass of the fainter, from i /6 to i /8 as great 
as the Sun's. The latter is the smallest mass so far established 
with any degree of probability for any star. 

The estimates of the magnitudes of the components vary 
considerably, but we may adopt 9.3 and 10.8 as approximately 
correct (Russell adopts 9.6 and 11.3). Now the Sun at the 
distance of Krueger 60 would shine as a star of the third mag- 
nitude, hence the two stars have actual luminosities only 
I /330 and I /1320 that of the Sun. According to Adams, the 
spectrum of the pair is of the Class Mb. This is undoubtedly 
the spectrum of the brighter star and the fainter one probably 
has a spectrum even more advanced. There is thus little 
question but that the mean density of each star is much 
greater than the Sun's and the intensity of its radiation per 
unit of surface area much smaller, and we may agree with 
Russell that these two stars are nearing "the very end of their 
evolutionary history." 

I may point out that our knowledge of this system is due 
primarily to the fact that it is, relatively speaking, so very near 
our own. The parallax of +0.256" corresponds to a distance 



232 THE BINARY STARS 

of about 12,7 light years. But the average star of the apparent 
ninth magnitude is more nearly 400 light years distant, say 
thirty times as far away. Remove Krueger 60 to such distance 
and its components become nearly 7>^ magnitudes fainter; 
that is 16.8 and 18.3 magnitude, respectively. The system 
would then be invisible as a double star in any existing tele- 
scope, and the probability of its detection on photographs 
taken with our giant reflectors would be extremely small. 

f CAPRICORNI AND 85 PEGASI 

Table I of Chapter VIII contains the orbits of ten systems 
which have revolution periods ranging from fifteen to twenty- 
seven years, as well as four of still shorter period, 5 Equulei, 
Ho 212, K Pegasi, and A 88. These systems have nearly all 
been discovered in recent years; only one, 42 Comae Berenices, 
dates back to Struve's time and one, 5 Equulei, to the time of 
Otto Struve; seven of them were discovered by Burnham. 

Several of these systems have already been referred to in 
more or less detail and it will suffice to describe briefly two 
others which are fairly typical of the group and which at the 
same time present some interesting contrasts. These are ^ 
Capricorni and 85 Pegasi. The former has a period of 21.17 
years, a small eccentricity, 0.185, and fairly high orbit inclina- 
tion, 69.4°. Since the major semi-axis is 0.565", the two com- 
ponents are well separated when at, or near, the extremities of 
the rather narrow apparent ellipse and are then easily meas- 
ured; but at minimum separation, when the angular distance 
is only 0.2", measures are very difficult, particularly from sta- 
tions in the northern hemisphere. As in all the binaries of this 
group, there is no apparent deviation from simple elliptic 
motion and the orbit elements are well determined. My orbit, 
computed in 1900, still represents the observed motion with 
precision although the stars have traversed an arc of nearly 
270° since then. The proper motion is small for so bright a 
star, less than 0.03" annually, and the parallax has not been 
determined. If we assume the mass to be twice that of the 
Sun, the parallax will be -fo.o6"; if the mass is eight times 
the solar mass, the parallax is +0.04". Probably these figures 



THE BINARY STARS 233 

may be regarded as approximate limits and we may therefore 
assume that each of the two components is from ten to twenty- 
five times as luminous as our Sun and that the orbit, in its 
dimensions, is comparable to Saturn's. The spectral class is 
A2, hence the stars probably exceed the Sun in surface bright- 
ness, but are probably also larger and less dense than the Sun. 

The orbit of 85 Pegasi is not quite so determinate as that of 
^ Sagittarii because the great difference in the magnitudes of 
the components makes measures difficult even when the ap- 
parent separation has its maximum value of about 0.8", and 
practically impossible near the time of periastron passage when 
the angular distance is only 0.25". Nevertheless, the orbit 
computed by Bowyer and Furner in 1906 represents the motion 
as observed to date within the limit of accidental error of 
measure, and we may regard the period at least as well deter- 
mined. Though the eccentricity, 046, and the orbit inclina- 
tion, 53.1°, are of average value, the fact that the line of nodes 
is nearly perpendicular to the major axis of the true orbit 
makes the apparent ellipse rather an open one. 

The system has the large proper motion of 1.3" annually, and 
according to Kapteyn and Weersma the parallax is +0.067''. 
Combining the latter value with the apparent magnitude, 5.8 
on the Harvard scale, we find that the bright star is almost 
precisely equal to the Sun in absolute magnitude ^ and since it 
h^s a spectrum of Class Go, we should expect the surface con- 
ditions in the two bodies to be similar. The spectrum of the 
companion is not known, but this star certainly radiates more 
feebly than its primary for it gives out less than i /lOO as much 
light (apparent magnitude ii.o). Now the interesting fact is 
that the independent investigations of Comstock, Bowyer and 
Furner, and Boss agree in making the smaller star from two to 
four times the more massive, though Boss adopts equal masses 
because he regards the meridian circle measures as of small 
weight and the result they give as a priori improbable. But 
we have in Sirius a system in which an even greater disparity 
between mass and luminosity in the two components is beyond 
question and it is by no means impossible that a similar rela- 
tion holds in the system of 85 Pegasi. The orbit elements and 



234 THE BINARY STARS 

parallax give a total mass 2.65 times that of the Sun, hence, if 
the brighter star is really the less massive of the two, its effec- 
tive radiating power must exceed that of the Sun though the 
two bodies give out light of the same spectral characteristics. 

e HYDRAE AND ^ CANCRI 

Reference has already been made to the multiple systems 
€ Hydrae and f Cancri and to the fact that there is a consid- 
erable number of triple, quadruple, and multiple systems 
among the stars known primarily as double stars. Such sys- 
tems raise many interesting questions ; as for example, whether 
it is possible to detect the influence of the more distant star or 
stars upon the orbital motion of the closer pair, or, conversely, 
the effect of the binary pair upon the observed motion of the 
other stars. 

As a matter of fact, it is a disturbing force of the latter kind 
that has actually been noticed in one of the two systems named, 
e Hydrae, while in f Cancri (and other pairs), an explanation 
for observed irregularities in the motion of the distant star has 
been found in the existence of an invisible fourth star. In each 
of these two systems the larger, or at least the brighter, star 
is the one which has been divided into a close binary, desig- 
nated for convenience as AB. The third star, C, in f Cancri 
has shown an annual relative motion of about 0.5° in an arc 
which is concave toward AB. The motion of C in e Hydrae [s 
of similar character, but slower; and in each case there are 
periodic irregularities in the motion such that when the ob- 
served positions are plotted they lie on a curve showing more 
or less definite loops at intervals of about eighteen years in 
the one case, and of fifteen years in the other. Seeliger's 
analysis leaves no question but that in the system of e Hydrae 
the irregularity is only apparent, being caused by the fact 
that the effective light center of the system AB describes a 
small ellipse by virtue of the orbital motion in this fifteen-year 
period binary. In the system of ^ Cancri, on the other hand, 
it seems to be real and to be due to the presence of a fourth 
star, invisible in the telescope, which revolves with C about a 
common center in a slightly eccentric orbit with a period of 



THE BINARY STARS 235 

17.6 years. In neither system, and, in fact, in no other visual 
triple, has it been possible to detect any disturbing effect, due 
to the more distant third star, in the orbital motion of the 
closer binary pair; but it must be remembered that the un- 
avoidable errors of observation are large in comparison with 
the possible perturbations. 

IJL HERCULIS AND 40 ERIDANI 

Frequently it is the smaller, or fainter star of a wide pair 
which is itself a close binary. Two of the most interesting 
systems of this kind are n Herculis and 40 Eridani. These are 
both bright stars, of magnitude 3.48 and 4.48, respectively, on 
the Harvard scale, with companions, noted as 9.5 and 9.2 by 
Struve, separated from their primaries by 32" and 82", respec- 
tively. Struve describes each of these wide pairs as "yellow 
and blue," and the color of the bright stars harmonizes with 
the spectral Class G5, assigned to them in the Revised Harvard 
Photometry. But, according to Adams, the 'blue' companion 
to M Herculis belongs to Class Mb, whereas the equally 'blue' 
companion to 40 Eridani belongs to Class A2 ! Evidently, the 
color contrast observed in such pairs is not a safe guide to 
difference of spectral class; and if it is not such in wide pairs 
like these, how much less is it to be trusted in closer pairs! 

The companion of ii Herculis was first noted as double by 
Alvan Clark in 1856, having escaped the search both of 
Herschel and of Struve. Since discovery, it has described more 
than a complete revolution, and the period and other orbit 
elements are quite definitely established, presenting no unusual 
features. The bright star, A, has an annual proper motion of 
0.817" in 203.35° and the binary pair, BC, is certainly moving 
through space with it, for the measures of AB since Struve's 
time show very little relative motion. Assuming that the 
parallax determined for A also applies to BC, I have given the 
mass of the binary, in Table IX of the preceding chapter, as 
just equal to that of the Sun. On the same assumption, the 
semi-major axis of the orbit has a length of twelve and one- 
quarter astronomical units, which is greater than that of 



236* THE BINARY STARS 

Saturn's orbit, while the eccentricity is about the same as that 
of the orbit of Mercury. 

The double companion to 40 Eridani forms a system drawn 
on a larger scale; its period is 180 years, so that since its dis- 
covery by the elder Herschel it has not had time to complete 
a full revolution. The elements, however, are fairly deter- 
minate and present the remarkable feature of the smallest 
eccentricity established in any visual binary. One would not 
expect to find this associated with a period of 180 years, but 
the pair is an easy one to measure with even moderately good 
telescopes and the measures since about 1850 are plentiful. 
The fact therefore seems to be beyond doubt. The bright star, 
A, has the exceptionally large proper motion of 4.1 1" annually 
in 213.3° and the faint pair is travelling with it, for as in the 
system of /z Herculis, the measures of AB indicate little relative 
motion. The comparative nearness to the Sun which would 
be inferred from this large proper motion has been confirmed 
by direct measures of the parallax. It is hardly possible to 
question the resulting mass and absolute magnitudes of 
the two components and yet it is most remarkable to find 
stars of such feeble luminosity belonging to the spectral 
Class A2. 

On the basis of the assumed orbit and parallax, the semi- 
major axis has a length of 27.5 astronomical units and the orbit 
is therefore nearly as large as Neptune's, while the binary is 
470 astronomical units from the bright star. This star is about 
seven-tenths of a magnitude fainter than the Sun would be if 
viewed from the same distance, but its spectral class is a little 
later, indicating somewhat feebler luminosity. We may there- 
fore assume that its mass is equal to the Sun's and that the 
period of revolution of the binary about the bright star will 
be about 7,000 years, if the present separation, 83", is the mean 
distance. However faulty these figures may be, there is no 
doubt at all but that in these triple systems we have repro- 
ductions on a vast scale of the Earth-Moon-Sun type of orbital 
motion, making due allowance for differences of relative mass 
in the components; indeed, we may expect to find systems of 
dimensions even greater than these. 



THE BINARY STARS 237 

This raises a question. The mere fact that two stars have 
the same motion through space is ordinarily held to be suffi- 
cient evidence of the binary character of any double star; but 
to what degree of apparent separation does this criterion hold 
good? Small stars are known which have the same proper 
motion as brighter ones 30' or more away,^ which, in the case 
of even moderate remoteness as stellar distances go, may 
correspond to an actual separation of half a light year or more. 
Certainly the two stars are physically connected and probably 
the}' have had a common origin; but does that imply orbital 
motion in the ordinary sense, or shall we simply say that they 
move through space along parallel paths as the stars in the 
Taurus cluster or those in the Ursae Major cluster do? This 
is one of the questions to which no general answer can be given 
at the present time. 

POLARIS 

A triple system quite different in type from those we have 
been discussing is the one of which the North Star, Polaris, is 
the only visible component. Six plates taken by Campbell 
with the Mills spectrograph in 1896, gave radial velocities for 
Polaris ranging only from —18.9 to —20.3 kilometers per 
second. As the plates were taken at varying intervals between 
September 8 and December 8, they seemed to furnish sufficient 
evidence of a constant velocity; but when additional plates 
were taken in August 1899, the first three gave velocities of 
— 13. 1, —II. 4, and —9.0 kilometers, respectively. "Inas- 
much," writes Campbell, "as a range of four kilometers is not 
permissible in the case of such an excellent spectrum, the star 
was suspected to be a short period variable," and plates were 
promptly secured on a number of additional nights. These 
settled the question, showing that the radial velocity has a 

iThe faint star (11. visual, 13.5 photographic magnitude) 2° 13' from a Centauri, 
for which Innes proposes the name Proxima Centauri, has practically the same proper 
motion and parallax as the bright star. The great angular separation, by reason of the 
exceptionally large parallax, corresponds to a linear separation which is only about twenty- 
two times that between 40 Eridani and its binary companion; but, even so, if the star is 
moving in an orbit with a Centauri, the period of revolution must be measured in hundreds 
of thousands of years! The large color index denotes extreme redness, and, intrinsically, 
it is by far the faintest star of which we have a definite knowledge. 



238 THE BINARY STARS 

range of six kilometers, the period of one complete oscillation 
being slightly less than four days. It appeared, upon investi- 
gation, that the six plates of 1896 were taken "at intervals 
differing but little from multiples of the period of the binary 
system and therefore fell near the same point in the velocity 
curve." 

The period 3.968+ days represents the observations in 1896 
and also the very numerous ones made from 1899 to the present 
time, and there is no evidence of any variation in this element 
nor in the value of K in the binary system. But the velocity 
of the center of mass for this system was about —17.2 kilo- 
meters in 1896.75, whereas it was only — 1 1.5 in 1899.75. Such 
a discrepancy could not possibly be attributed to errors of 
observation or measurement and were rightly regarded by 
Campbell as clear evidence that Polaris is at least a triple 
system, the four-day period binary moving in a much larger 
orbit with a third star, invisible to us. At one time it was 
thought that this larger orbit had a period of about twelve 
years, but this is not the case. The maximum velocity (min- 
imum negative velocity) in this orbit seems to have been 
reached in 1899 or 1900; in 1910.5 it was about —15.8, in 
1916.2, —17.8 kilometers. What the minimum value (maxi- 
mum negative velocity) will be cannot now be predicted, but 
it is apparent that the orbit is quite eccentric with a 'period 
exceeding twenty years. The ratio of more than 1,800 to i 
shown by the long period oscillation to that of short period is far 
greater than the ratio of the periods in any known visual triple. 

Campbell has described the spectrum as 'excellent* ; exami- 
nation of several of the plates shows that the absorption lines 
are numerous and well defined. They are not broad nor yet 
hazy, but compare very favorably with the lines in the solar 
spectrum as seen in the light reflected by Venus, and the prob- 
able error of measure from a single plate is therefore less than 
half a kilometer. Though the range in velocity is small, the 
character of the spectrum places it beyond doubt; and that 
this variation is due to orbital motion in a binary system was 
until quite recently questioned by no one, for at the time of its 
discovery and for many years thereafter Polaris was regarded 



I 



THE BINARY STARS 239 

as "perhaps the star in all the sky of whose constancy in light 
we may be most certain." Indeed, it had been adopted by 
Pickering as the standard star (magnitude 2.12 in the Revised 
Harvard Photometry) in the extensive photometric work car- 
ried out at Harvard College Observatory, and had held the 
same position in similar researches elsewhere. 

In 191 1, however, Hertzsprung was able to show that it was 
really a variable star, the light curve resembling, in general, 
that of the Cepheid variables. The period was found to be 
identical with that of the velocity variation, the range in light 
(photographic), 0.17 magnitudes. Examination of the exten- 
sive photometric data at Harvard confirmed the discovery, as 
did the observations made elsewhere, and it was shown, as 
Hertzsprung had anticipated, that the range of light, visually, 
is only about one-tenth of a magnitude, thus adding another 
point of resemblance to the Cepheid variables which, as a class, 
show a greater range of light variation in the light of short 
wave-length than in the light which most strongly affects 
the eye. 

Now the spectrum, and the characteristics of the four-day 
orbit of Polaris had already been recognized as strikingly sim- 
ilar to those of the known Cepheids; in fact, this was one of 
the reasons for Hertzsprung's investigation. We must cer- 
tainly, therefore, class Polaris among the Cepheids and must 
face the question whether we shall give up the well-founded 
belief that it is a short period binary system simply because 
we now discover that its light varies in a particular manner in 
the same period as its velocity. Frankly conceding that no 
theory so far advanced for the cause of the light variation, on 
the assumption that it is somehow due to the interaction of 
the two components in a binary system, is wholly acceptable, 
I am still of opinion that we have no reason to abandon our 
faith that it is a binary until some substitute theory is brought 
forward which will account for the periodic displacement of the 
spectral lines. All theories so far advanced, on the hypothesis 
that in a Cepheid variable we are dealing with physical changes 
in the atmosphere of a single star, fail to explain this line dis- 
placement. Moreover, they call for broadened, and probably 



240 THE BINARY STARS 

hazy absorption lines instead of the well-defined and quite 
narrow lines which characterize the spectra of Polaris and 
many other Cepheids. 

CASTOR 

The spectroscopic binaries ai and a2 Geminontm, which form, 
respectively, the fainter and the brighter component of the 
well-known double star, Castor, present an interesting contrast 
in the forms of their orbits. Curtis's definitive investigation 
shows that the orbit of ai (discovered by Belopolsky in 1896, 
period 2.928285 days) is practically circular, the rigorous least 
squares solution giving the eccentricity o.or; but the orbit of 
a2 (discovered by Curtis in 1904, period 9.218826 days) has an 
eccentricity 0.5033, above the average value for the visual 
binaries. The one, then, is typical, the other exceptional. 

Unfortunately, the elements of the visual binary cannot yet 
be regarded as determinate though Castor was, as we have seen, 
the first stellar system for which orbital motion was definitely 
established. The latest and most thorough research relating 
to this orbit is the one by the veteran computer, W. A. Do- 
berck, the man who has investigated more double star orbits 
than any other astronomer. He gives three alternative sets of 
elements with periods respectively, 268,347, and 501 years, but 
regards the 347-year period as the most probable. Recent ob- 
servations seem to support this conclusion, and Curtis has 
adopted it in his speculations concerning the system. The 
relative radial velocity of the two visual components, derived 
from the latter's investigation of the spectroscopic binary 
orbits was 7.14 kilometers. This, with Doberck's elements, 
gives a parallax of +0.05" and a total mass 12.7 times that of 
the Sun. Moreover, if the two spectroscopic binaries are re- 
volving in the same plane as the visual system, for which 
Doberck finds the inclination, 63.6°, the semi-major axes of 
the two systems are: 

ai Gemi7wnim, a = 1435,000 kilometers 
a2 Geminorum, a = 1,667,000 kilometers 

that is, they are of the same order of magnitude. Therefore, 
to account for the relative periods, it is necessary to assign to 



THE BINARY STARS 241 

the fainter star a mass about six times as great as that of the 
brighter one. Finally, on the generally accepted theories of 
stellar evolution, the difference of eccentricity means that the 
brighter and less massive system is the older; the fainter sys- 
tem, with circular orbit, comparatively, of recent origin. These 
anomalous results are, of course, at present almost entirely 
speculative. Curtis finds no evidence of irregularities in the 
velocity curves, nor of light variation in cither component of 
the visual pair. 

5 ORIONIS 

An inspection of the elements of 5 Ononis would not lead to 
the Impression that it was distinguished in any particular 
manner from the other short period spectroscopic binaries 
listed in Table II. It has, however, several points of interest. 
The faintest of the three stars in the Belt of Orion, it is one of 
the spectroscopic binaries Investigated by Stebblns with the 
selenium photometer, the measures definitely establishing the 
fact that it is a variable star with a light range of 0.15 magni- 
tudes of which 0.08 magnitudes is due to eclipses. The two 
stars of the system are probably ellipsoidal in form and keep 
always the same face turned toward each other. In other 
words, under the action of powerful tidal forces the rotation 
period of each has been brought to, or kept in equality with 
the period of its orbital revolution. The light curve also indi- 
cates that the surface brightness of the disks is not uniform, 
each body being "brighter on the front side In its motion in 
the orbit." The mean mass of the system is determined as 
0.006 that of the Sun, the larger star having a radius at least 
five, and the smaller one a radius at least 1.4 times that of 
the Sun. 

It Is, however, a discovery made by Hartmann from the 
observations of the star's spectrum at Potsdam in the years 
1900 to 1903 that gives the system its peculiar position among 
spectroscopic binaries. The spectrum Is of Class B and the 
calcium lines (known as the H and K lines) are narrow and 
sharply defined, while the other lines, chiefly due to helium and 
hydrogen, are more or less diffuse. Now Hartmann 's measures 



242 THE BINARY STARS 

of the hydrogen and helium lines indicated a range in radial 
velocity of about 200 kilometers per second, but those of the 
calcium lines gave a nearly constant velocity! From a series 
of plates taken at Allegheny in the years 1908 to 191 2, Jordan 
has arrived at similar results, the H and K lines giving a 
nearly constant velocity whose mean is +18.7 kilometers 
whereas the range from the other lines is almost precisely 
200 kilometers. 

The velocity of the center of mass of the system, as derived 
by Jordan from lines not due to calcium is 4-15-2 kilometers, 
diflfering from the mean for the calcium lines by 3.5 kilometers, 
and Hartmann's measures indicated an even greater discrep- 
ancy, 7 kilometers. This may be evidence that the material 
producing the calcium absorption does not belong to the star.^ 
Take into account two other facts, (i) that the constellation 
of Orion is well known as a region of space containing wide- 
spread nebulosity, and (2) that the Sun's own motion through 
space is carrying it away from that region with a velocity of 
about 18 kilometers per second, and it is apparent that Hart- 
mann's assumption of the existence between us and 8 Ononis 
of a cloud of calcium vapor stationary in space (so far as radial 
velocity is concerned) has much to commend it. It is entirely 
possible that the explanation might have won general accept- 
ance if 5 Ononis had remained the only star showing this 
anomaly. But the number has gradually increased until now 
some twenty or twenty-five binary systems are known in 
which the H and K lines yield either constant velocities or 
velocities which have a different range — generally in the sense 
of being much smaller — from that derived from the other lines 
of the spectrum. 

Every one of these stars is a 'helium star\ that is one belonging 
to Spectral Classes Oe to B2; and many astronomers have 
asked the question pertinently raised by Young, "Why should 
the calcium clouds always lie in front of a star of type B2 or 
earlier?" In his paper on the orbit of x Aurigae, a star for 
which the H and K lines give a velocity range approximately 

» It is not impossible that the discrepancy is due, in part at least, to errors in the wave- 
lengths assumed for some of the standard lines. 



THE BINARY STARS 243 

half as great as that for the other lines, Young summed up the 
other known facts relating to systems of this type. Omitting 
details and specific illustrations, these are essentially as follows: 

In eight stars the calcium lines give a velocity that remains 
nearly constant and that differs somewhat, as a rule, from the 
velocity of the system determined from the other lines which 
in every instance show a large range in velocity. 

"In several stars the calcium lines are known to vary differ- 
ently from the other lines." Orbits have been computed from 
the H and K lines for four such stars, in two instances with 
amplitudes about half those given by the remaining lines. 
Young believes that future investigation will reveal all grada- 
tions in the variation of the calcium lines from constant veloc- 
ity to oscillations equal to those of the other lines in the 
spectra; and, indeed, several stars are already known in which 
there is not much difference in the mean velocities from the 
calcium and from the other lines. 

"There seems to be no exception to the rule that when the 
calcium lines are sharp and narrow and the other lines broad, 
the star exhibits a variable radial velocity." This relation was 
first announced, I believe, by Frost, who has used it in pre- 
dicting, with success, that certain stars with spectra of the 
character described would prove to be spectroscopic binaries. 

It is very difficult to harmonize all of these facts with the 
theory of stationary calcium clouds independent of the systems, 
and therefore at least two other theories have been advanced. 
One of these, that the phenomena are due to anomalous dis- 
persion effects may be passed with the mere statement that it 
has not won wide acceptance. The other, that the calcium vapor 
in question envelops one or both stars, lying high above the 
effective photosphere, is far more plausible. So far as I am 
aware, this theory was first put in definite form by Lee in his 
discussion of the orbit of 9 Camelopardalis. He shows, as does 
Young in the paper already cited, that it will not only account 
for the phenomena observed in the particular system under 
investigation, but that it is sufficiently elastic to meet the 
varying demands made upon it by conditions in other systems. 
However, the theory at best is as yet only a working hypothe- 



244 THE BINARY STARS 

sis, which must meet the tests of many further applications 
before it can be regarded as definitely established. 

/3 AURIGAE 

The binaries which imprint the spectra of both components 
upon the plates are of special interest, because the difference 
in the range of oscillation shown by the two sets of lines permits 
us to determine the mass ratio, while their intensities afford 
us a measure of the relative brightness of the two components. 
That there is a close correspondence between difference of 
mass and difference of brightness is a relation to which Schle- 
singer. Baker and others have called attention. 

In connection with Table VIII of the preceding chapter, in 
which are listed all systems for which I could find published 
values of the mass ratio, I commented upon the fact that 
nearly all of these stars belong to Spectral Classes B and A. 
In his paper on a Virginis, in 1909, Baker made the comment 
that, as a rule, they also have short periods and nearly cir- 
cular orbits. If the stars in Table VIII are examined with 
respect to these two elements, it will be found that two-thirds 
of them have periods under ten days, and that two-thirds (not 
always the same stars) have eccentricities under 0.08. 

One of the most interesting stars of this class is jS Aurigae, 
the second spectroscopic binary star in point of discovery. The 
two components are of nearly equal brightness and have iden- 
tical spectra, of Class A. Baker defines the spectra more pre- 
cisely by saying that "they are further advanced than that of 
Sirius and are intermediate to those of the components of 
Castor." The magnitude of the star on the Harvard scale is 
2.07, hence each component is about 2.8 magnitude; which 
means, if we adopt Kapteyn and Weersma's parallax, 0.014", 
that each is about 6>^ magnitudes brighter than our Sun 
would be if viewed from the same distance. Their radiating 
power per unit of surface area doubtless greatly exceeds that 
of the Sun, but even so they must be vastly larger than the Sun 
and probably much less dense. 

The system has been the subject of extensive investigations by 
the astronomers at the Harvard, Potsdam and Pulkowa Obser- 



THE BINARY STARS 245 

vatories, but the most recent and most complete discussion is 
Baker's, based primarily upon the spectrograms secured at the 
Allegheny Observatory in 1 908-1 909 but utilizing also the results 
of the earlier researches. From Baker's definitive elements it 
appears that the two masses are 2.2i/sin'z and 2. i7/sin*t, 
the Sun's mass being the unit, and that the mean distances 
of the two components from the center of mass of the 
system are respectively 5,934,000/sini, and 6,047,000/sini 
kilometers. The value of i, the inclination, is of course un- 
known, but in view of the range of 220 kilometers in the radial 
velocity it is almost certainly as great as 30°; and since sin 
30° = 0.5 we may say that the superior limit to the two masses 
is probably about seventeen times the Sun's mass, and to the 
linear distance between the centers of the components, not 
more than 24,000,000 kilometers. The actual values may be 
decidedly smaller. Now 24,000,000 kilometers is less than 
one-sixth the distance from the Earth to the Sun, hence it is 
evident, when we recall the parallax given above, that no 
telescope can show the system as a double star; even were 
the inclination as low as 6°, the angular separation would be 
barely 0.0 1". 

Baker's analysis of the Allegheny observations gave strong 
evidence of systematic departures from elliptic motion which 
could be represented as a secondary oscillation with a period 
one-third that of the primary. This is a phenomenon fre- 
quently noted in spectroscopic binary star orbits and has 
received special attention at the Allegheny Observatory. 
Schlesinger has shown that it is sometimes merely a "blend 
effect" due to the unrecognized presence of a faint spectrum 
produced by the companion star. This broadens the lines 
with the result that the measures give velocities displaced at 
any given point toward smaller values. In |8 Aurigae no such 
effect can operate for the orbit "is derived exclusively from 
measures of the separated component lines." A comparative 
study of the Potsdam, Pulkowa and Allegheny measures of 
velocity shows, however, that the secondary oscillation has no 
physical basis, that is, that it is not due to any perturbation in 
the stars' motions, for the range of the oscillation varies with 



246 THE BINARY STARS 

the instrument employed, being "greatest for the spectrograph 
giving the least dispersion and the smallest separation of the 
two spectra." There are doubtless cases in which such oscil- 
lations have a physical basis, as I have remarked on an earlier 
page, but the fact that in jS Aurigae, in 30 H Ursae Majoris and 
in other systems subjected to a searching analysis the oscilla- 
tions have been found to lie in the measures rather than in the 
star's motion warns us against accepting the reality of such 
appearances until the most thorough tests have been applied. 
One of the most striking results developed by Baker's dis- 
cussion is that the period of revolution is apparently slowly 
lengthening. Intercomparison of the observations at Potsdam 
in 1 888-1 897 and 1 903-1 904, at Pulkowa in 1 902-1 903 and at 
Allegheny in 1908- 1909, shows a progression from 3.95993 
days for the epoch 1896.4 to 3.960029 days for the epoch 1906. i , 
an increment of -fo.ooooio days, or +0.86 seconds annually. 
Similar variations in period, always in the sense of a slow in- 
crease with the time, have been found in three other spectro- 
scopic binaries which are not eclipsing variables; in e Herculis 
by Harper, in X Ayidromedae by Burns, and in 6 Aquilae by 
Baker. Granting the uncertainties that may attach to the 
numerical values and even to the fact of the increase in one 
or two of the systems, the results as they stand lead Baker to 
the query, "Are we not here actually observing the progress of 
evolution from the spectroscopic to the visual binary?" This 
is a point of the greatest interest, but we must regard the query 
as speculative only until further observations have established 
the fact of progressive increase of period beyond question. If 
a series of spectrograms of these stars taken at the present 
time, and another series taken, say, five or ten years later, 
show that the progression continues, we may accept it as strong 
evidence of a definite advance in the evolution of the stars. 

ALGOL 

j8 Persei, better known as Algol, has been the subject of 
many memoirs which are counted among the classics of astro- 
nomical literature, but our knowledge of the mechanism of 
the system is still far from satisfactory. The general character 



THE BINARY STARS 247 

of the light variation, it is true, was established by Goodericke, 
its discoverer, as early as 1783, and his hypothesis as to its 
cause has been fully confirmed by the spectrograph ; but even 
now we cannot regard the light curve as definitively estab- 
lished, and as measures of the radial velocity accumulate, we 
find ever further evidence of complexities in the system. An 
adequate account of the work that has been done on this star 
and of the theories that have been offered in explanation 
of the observed phenomena would require a chapter. The 
present note must be limited to a description of some of 
the more striking facts which have been developed in these 
researches. 

Argelander first demonstrated the existence of fluctuations 
in the period between successive light minima, and Chandler's 
more extensive studies, utilizing all available observations 
from Goodericke's time to 1888, not only confirmed this con- 
clusion but led him to explain them as arising from a long 
period inequality which he ascribed to the presence of a third 
body in the system revolving with the eclipsing binary in a 
practically circular orbit in a period of about 130 years. 
Chandler's theory predicted a slow increase in the period of 
the eclipsing binary {i. e., in the length of the interval between 
successive light minima) beginning with the closing years of the 
Nineteenth Century and continuing until late in the Twentieth, 
and this is apparently supported by Stebbins whose period for 
the variable, derived from observations made in 1909-1910, is 
six seconds longer than Chandler's for the years 1 871 to 1888. 
Chandler, moreover, supported his hypothesis by an analysis 
of the proper motion of Algol, finding evidence of variations 
similar in character (but of much longer period) to those 
which led to Bessel's predictions with respect to Sirius and 
Procyon. 

While the depth and duration of the primary minimum (loss 
of light equals about 1.2 magnitudes, length of eclipse, from 
first contact to fourth, about ten hours of the 68.8 hour period) 
had been defined with satisfactory precision at least as long 
ago as Schonf eld's time, no decisive evidence of variation in 
the normal brightness between these minima was forthcoming 



248 THE BINARY STARS 

until 1 909-1 9 10. Stebbins's measures with the selenium cell 
photometer then established the existence of a secondary min- 
imum of only 0.06 magnitude, and also indicated that the light 
is not strictly constant at any phase, the maximum brilliancy 
falling not half way between the two minima, as we might 
expect, but just before and just after the secondary minimum. 
While Stebbins's measures are of remarkable accuracy, he 
would be the first to say that it is highly desirable to have his 
results, especially for the minute variation between minima, 
confirmed by additional measures with the new photo-electric 
cell photometers. 

Pickering, in 1880, had worked out the dimensions of the 
system in terms of the (unknown) linear separation of the two 
components, on the eclipse hypothesis; Vogel, in 1889, not 
only put this hypothesis beyond question by his demonstration 
that the star is a spectroscopic binary, but also from his meas- 
ures of the radial velocities determined the diameters of the 
two stars, as well as the dimensions of the orbit, in terms of 
kilometers, making the assumption, however, that the two 
components are of equal mass. At the time, Vogel's work 
seemed to leave little to be desired, but in more recent years 
Schlesinger and Curtiss have found that the light minima lag 
from one and one-half to two hours behind the time demanded 
by the spectrographic measures on the eclipse theory. The 
ephemeris based upon Stebbins's more accurate light curve, 
it is true, has removed i h. 16 m. of this discrepancy; but 
since the Allegheny observers find a similar lag in the case of 
several other Algol variables it must still be regarded as a 
matter demanding investigation. 

In 1906, Belopolsky announced a long-period oscillation 
in the radial velocities of Algol, and this was confirmed by 
Schlesinger in 1912. The latter finds the period to be 1.874 
years, the semi-amplitude (K) 9.14 kilometers, and the orbit 
nearly circular, and his examination of the photometric ma- 
terial from 1852 to 1887 reveals evidence of a corresponding 
oscillation in the times of light minimum. 

Inadequate as this summary is, it is yet sufficient to show 
how great is the complexity of the system, and to indicate the 



THE BINARY STARS 249 

necessity for further accurate photometric and spectrograph ic 
measures for the formulation of a complete theory. So far as 
the eclipsing binary is concerned, we can, however, give a 
description of the principal features that is fairly reliable. The 
following account is based upon Stebbins's results, but whereas 
he gives figures for several alternative assumptions, I shall 
give only those which seem to me to be nearest to the truth. 
The companion to Algol probably rotates upon its axis once 
in every revolution and thus, like our Moon, keeps the same 
face turned toward its primary, and this face is brighter than 
the opposite side by reason of the radiation received from 
Algol. The primary may also rotate once in a revolution but 
of this we have no evidence. Possibly the two bodies are 
slightly ellipsoidal in form and, as Shapley suggests, may pos- 
sess more or less extensive absorbing atmospheres which pro- 
duce a gradual darkening of the disks toward the edge, or limb. 
Assuming the stars to be spherical and without such absorbing 
atmospheres, the light curve gives 82.3° for the inclination of 
the orbit plane. It is a reasonable assumption, also, in view 
of their relative brightness, that Algol is twice as massive as its 
companion. If we adopt this ratio and the value 1,600,000 
kilometers for a sin i, we have the following figures for the 
dimensions, masses and densities, the Sun being taken as 
the unit: 

Radius of Algol = i .45 

Radius of companion = i .66 

Mass of Algol = 0.2,7 

Mass of companion = o. 18 

Density of Algol = o. 12 

Density of companion = o. 04 

The distance between the centers of the two stars is 4.77 
times the radius of Algol and the mean density of the system 
0.07 that of the Sun. 

The stellar magnitude of Algol is 2.2; the light of the faint 
hemisphere of the companion equals that of a star of 5.2 mag- 
nitude, the light of the brighter hemisphere that of a star of 
4.6 magnitude. If the value +0.029", ad^ted for the parallax 
by Kapteyn and Weersma, is correct, the light of Algol is 



250 THE BINARY STARS 

i6o times that of the Sun, the light of the faint and bright 
hemispheres of the companion, respectively ten and seventeen 
times that of the Sun. If the value +0.07" adopted in some 
earlier discussions of the system is correct, the luminosities 
are reduced to about one-sixth of those just given. We may 
safely regard the latter as the minimum limits and may 
therefore say that even the faint side of the 'dark' companion 
to Algol is far more brilliant than our Sun. 

It was inevitable that in this chapter more emphasis should 
be placed upon the systems in whose study we encounter diffi- 
culties than upon those which apparently conform to the laws 
of motion in a simple Keplerian ellipse. The chapter might be 
extended indefinitely; systems like 70 Ophiuchi, 4> Persei, 
j8 Lyrae, to name no more, present problems and puzzles which 
are still unsolved. It is precisely the unexplained anomalies, 
the irregularities in the motions, the apparent contradictions 
among the spectral lines, the complexities of various kinds, 
that oflfer the best opportunities for discoveries which may 
advance our knowledge of the forces at work in the stellar 
systems. If it were not for these, our interest in the systems 
themselves would soon flag; but as the case stands there is 
always some new problem to spur us on, "for each stellar sys- 
tem," as Miss Gierke says, "is in effect a world by itself, 
original in its design, varied in its relationships, teeming with 
details of high significance." 

An excellent illustration is found in the system of k Pegasi^ 
now under investigation at the Lick Observatory by Dr. F. 
Henroteau. Dr. Campbell, in 1900, discovered that one of the 
components of the well known 11.3-year period visual binary 
was itself a binary with a period of about six days. It now 
develops that the spectrum of the other component in the visual 
system is also present upon the plates, thus making it possible 
to determine the parallax, the mass of the system, the true 
inclination of the orbit plane, the linear dimensions of the 
triple system, the relative masses of the visual components, 
and an independent orbit of the 1 1 .3-year period binary. Com- 
parison of the plates taken in the present year with the earlier 



THE BINARY STARS 25I 

ones already shows the variation of the velocity of the center 
of mass of the six-day period binary, due to its motion in the 
larger orbit, and gives strong indications of a revolution of the 
line of apsides, a perturbative effect suspected in several other 
short period spectroscopic binaries, but not, so far as I am 
aware, established beyond question in any case.^ 

REFERENCES 

It is imnecessary to give a list of the many papers which have 
been consulted in .the preparation of this chapter. The refer- 
ences in earlier chapters will suffice for the visual systems and 
Campbell's 'Third Catalogue of Spectroscopic Binary Stars', 
now in preparation, will give the references for all spectroscopic 
binaries. 

• Dr. Henrotcau's investigation of the system of k Pegasi has been published as Lick 
Observatory Bulletin, Number 304. 



CHAPTER X 

A STATISTICAL STUDY OF THE VISUAL DOUBLE 
STARS IN THE NORTHERN SKY 

The Lick Observatory double star survey referred to in my 
historical sketch was undertaken with the definite purpose of 
accumulating data for the statistical study of the number and 
distribution of the visual double stars. I sought answers to 
such questions as these: What is the number of double stars 
relatively to the number of all stars to a given magnitude? 
Is the ratio the same for faint stars as for bright ones, in one 
part of the sky as in another, for stars of one spectral class as 
for stars of other classes? The answers which the survey 
affords for the stars in the northern half of the sky will be 
considered in the present chapter. 

The data consist of all visual double stars as bright as 
9.0 B. D. magnitude which fall within the distance limits set 
by the following 'working definition' of a double star proposed 
by me in 191 1 : 

(i) Two stars shall be considered to constitute a double star when 

the apparent distance between them falls within the following limits: 

l" if the combined magnitude of the components is fainter than 11 .0 

3" if the combined magnitude of the components is fainter than 

9.0 B.D. 
5" if the combined magnitude of the components lies between 6.0 

and 9.0 B. D. 
10" if the combined magnitude of the components lies between 4.0 

and 6.0 B. D. 
20" if the combined magnitude of the components lies between 2.0 

and 4.0 B. D. 
40" if the combined magnitude of the components is brighter than 
2.0 B.D. 

(2) Pairs which exceed these limits shall be entitled to the name 
double star only when it has been shown (a) that orbital motion exists: 



THE BINARY STARS 253 

(b) that the two components have a well defined common proper mo- 
tion, or proper motions of the 61 Cygni type; (c) that the parallax is 
decidedly greater than the average for stars of corresponding magnitude.* 

In all, there are 5,400 pairs, distributed according to the 
discoverer as follows : 



W. Struve 


1053 


0. Struve 


296 


Burnham 


551 


Hough 


237 


Hussey 


766 


Aitken 


2057 


Miscellaneous 


440 



The chief contributors to the 'miscellaneous' list are Jonck- 
heere, Espin and observers of the Astronomische Gesellschaft 
Star Catalogue. Sir John Herschel's long lists contribute only 
twenty-eight pairs and Sir William Herschel's discoveries are in- 
cluded in the number credited to W. Struve. It should be noted 
that a given system is counted only once though it may have 
three or even four or more components. In such triple systems 
as ^ Cancri, 7 Andromedae, and € Hydrae, the closer pair is 
listed ; in an occasional system in which the close pair is very 
faint the wider pair is the one counted. A number of the stars 
credited to Burnham and later observers also have S or oS 
numbers which are disregarded in the above tabulation. 

The first question to consider is whether the data are homo- 
geneous, for it is obvious that they can make no claim to be 
exhaustive. However carefully an observer may work, some 
pairs which he might discover with a given telescope will 
surely escape him. His eye may be fatigued, unnoticed haze 
or momentary bad seeing may blur out a faint companion star, 

> The definition, with correspondence relating to it, will be found in the Astronomische 
Nachrichlen {188, 281, 1911). Comstock and E. C. Pickering there suggest limits based 

upon the apparent magnitude, the former using the formula s=c (~z~) "*. the latter, the for- 
mula, log. 5 — c — 0.2 m, where 5 is the distance in seconds of arc between the components, 
m. the apparent magnitude, and c an arbitrary constant. If the values of c in the two 
formulae are so chosen as to give the limit 5.0' for stars of magnitude 6. 0, the formulae 
will give the limits 0.75' and 1.25', respectively, for stars of 90 magnitude. From the 
theoretical point of view either formula gives more logical limits than the ones in my 
definition, but there were practical considerations, fully stated in the article referred to, 
which led to the adoption of the latter. 



254 THE BINARY STARS 

or it may chance that at the date of examination the two com- 
ponents are so nearly in conjunction as to be below the resolv- 
ing power of the telescope. The number of known double 
stars can only be regarded as the lower limit to the number 
which might be discovered. Homogeneity was earnestly 
sought for, care being taken to work only when in good phys- 
ical condition and when the seeing was good, the practical test 
being the power to recognize very close and difficult pairs at 
a glance. But variations in the conditions are inevitable when 
the working program requires years for its execution and 
doubtless such variations have affected the present results. 
Careful comparison, however, shows no discernible difference 
in the thoroughness of the work done at different seasons of 
the year or in different parts of the sky, and it may fairly be 
said that the results of the survey represent the capacity of 
the combination of telescope and observer under average good 
atmospheric conditions at Mount Hamilton. If the work had 
all been done with the thirty-six-inch refractor the resulting 
data might be considered quite homogeneous. Unfortunately, 
a considerable part of it, including practically the entire area 
north of +60° declination, was done with the twelve-inch 
telescope, and it becomes necessary to consider the relative 
efficiency of the two instruments. 

I have applied two tests : first, the comparison of the most 
difficult pairs discovered with each instrument; second, the re- 
examination with the thirty-six-inch of some 1,200 stars pre- 
viously examined with the twelve-inch telescope. I find that, 
under the usual observing conditions, a pair with nearly equal 
components separated by only o. 15'', or a companion star as 
faint as 14.5 magnitude and not less than 1.5" from its pri- 
mary is practically certain of detection with the thirty-six- 
inch; with the twelve-inch, the corresponding limits in the 
two cases are 0.25* and 13 to 13.5 magnitude. Twelve new 
double stars were added by the re-examination of the 1,200 
stars. From these tests, taking into account the proportion 
of the whole work done with the twelve-inch telescope, I con- 
clude that about 250 pairs would have been added if the entire 
northern sky had been surveyed with the thirty-six-inch. 



THE BINARY STARS 



255 



According to Seeliger's count of the B. D. stars there are 
100,979 as bright as 9.0 magnitude in the northern hemis- 
phere. Of these, 5,400, or i in 18.7 on the average, have 
actually been found to be double within the limits set above. 
If we add only 200 pairs, the ratio becomes i : 18.03. A 
definite answer is thus given to my first question: "At least 
one in every eighteen, on the average, of the stars in the northern 
half of the sky which are as bright as g.o B. D. magnitude is a 
close double star visible with the thirty-six-inch refractor." There 
is no reason to doubt that the ratio is equally high in the south- 
ern half of the sky. 

TABLE I 

The Distribution of Double Stars in Right Ascension and 
Declination 





oh 


2h 


4h 


6h 


8h 


loh 


I2h 


I4h 


I6h 


I8h 


20h 


22h 


R.A. 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 




ih 


3»^ 


5^ 


7h 


Qh 


Ilh 


i3»» 


IS*^ 


I7h 


IQh 


2lh 


23h 


Decl. 


























o-'-Q^ 


6.3 


6.3 


7-4 


6.0 


6.0 


5-9 


5.5 


6.4 


5.4 


6.0 


5-3 


4-9 


10-19 


5-4 


5-4 


6.4 


TT 


5 


5 


5 


5 


4 


8 


6.1 


4 


5 


6.0 


5 


4 


5 


2 


20-29 


5-2 

6.0 


5.6 

5.5 


5-9 


5.1 

4-4 


4 
4 


8 
3 


6 
4 


4 

8 


5 
5 


2 

4 


4.8 
41 


5 

5 


4 



6.2 


4 
5 


9 

2 


4 
4 


2 


30-39 




8 


40-49 


6.7 
6.2 


A^ 


FT 

8.0 


4-5 
5-2 


4 
4 


5 
8 


6 
3 


2 
6 


3 
4 


2 

4 


5.6 
6.0 


5 
4 


5 
6 


4.2 

4-4 


6 


2 

8 


4 
4 


9 


50-59 


TT 


9 


60-69 


J± 


50 

3.8 


71 
4.2 


2.9 

3-5 


4 
3 


3 

I 


4 
2 


4 
6 


2 
2 


6 

8 


4.1 
2.2 


3 
3 


6 
4 


41 
4-9 


5 
5 


3 



6 


^ 


70-79 


5-9 


T 


80-89 


(oh to 5h) 


3.7 


(6h tOllh) 


3 


3 


(l2htOI7h) 


3 


4 


(I8ht023h) 


3 


9 



The figures give the percentages of double stars among stars to 9. o B. D. 
magnitude; the average percentage for the whole northern sky is 5.35. 

Table I exhibits the distribution of the 5,400 double stars in 
right ascension and declination as compared with the distribu- 
tion of the B. D. stars to 9.0 magnitude, the figures giving the 
percentage of double stars in each area. There are obvious 
irregularities in the table but no evidence of systematic differ- 
ences that can be regarded as seasonal effects. The percentages 
are as high in the sky areas surveyed in winter as in those sur- 
veyed in summer. There is a falling off in the percentage in 



256 



THE BINARY STARS 



the high declinations, especially in the regions well removed 
from the Milky Way, which is doubtless due in part to the 
fact that the area north of 60° was almost entirely surveyed 
with the twelve-inch telescope. The broken line in the table 
represents very roughly the position of the central line of the 
Milky Way, and it will be noted that the percentages near 
this line are, in general, above the average. 

^The distribution with respect to the plane of the Milky Way 
is more clearly brought out when the stars are tabulated ac- 
cording to galactic latitude. This has been done in Tables II 
and III, in the former of which the stars are divided in classes 
according to magnitude and the latitudes into zones each 20° 

TABLE II 

The Distribution of Double Stars by Magnitude Classes and 
Zones of Galactic Latitude 





m 


m 


m 


m 


m 


m 




Mag. to 


6.5 


6.6-7.0 


7.1-7.5 


7.6-8.0 


8.1-8.5 


8.6-9.0 


Total 


Zone 
















I 


19 


13 


14 


29 


40 


84 


199 


II 


43 


28 


50 


68 


114 


193 


496 


III 


60 


43 


56 


79 


148 


254 


640 


IV 


96 


54 


81 


132 


232 


401 


996 


V 


121 


88 


133 


249 


376 


653 


1,620 


VI 


84 


51 


81 


134 


221 


395 


966 


VII 


28 


23 


18 


54 


90 


154 


367 


VIII 


7 


6 


5 


12 


31 


55 


116 


Total 


458 


306 


438 


757 


1,252 


2,189 


5,400 



wide, beginning at the north galactic pole. Zone V therefore 
includes the area from +10° to —10° galactic latitude, and 
Zone IX, which ends at the south galactic pole and lies entirely 
below the equator, is not represented. As was to be expected, 
the numbers in every column of this table are largest in Zone V 
and fall to minima in Zones I and VIII. The question is 
whether this condensation toward the Milky Way is greater 
than that of all the stars. Table III provides the answer. 



THE BINARY STARS 



257 






I 
•*^ CO 



cS 






6 



1 



Os 



8 ^J 



^ 



CO g 










00 


7) 

d 


»0 r^ r*! Q 

n CO t^ HH 

fO rO CO vO 






M. 






Q 


n rj- Tj- 00 Q 
00 00 n 6 
tc T^ Tj- t>. 


ov r^ t^ 

00 CI M 

t^ 10 10 




H* 








• 00 
00 




5 8.5 ^8 

c< CO ro vO 


-+ t^ to 

\0 -rt- ■^ 




H^ 








ON M c* Q 

^ ^ s=> ?l 8 


ON f5 VO 

r^ lo Tf 




« 








00 

1 





»o cs 

rO rO CO »0 


ON »0 
rO J>- 10 

fO <S 











d 


^ rjr 
C4 ro 
Tj- Tl- 10 t^ 


M ro 
1^ 00 r^ 
r^ ■^ CO 




HH 






10 


d 


vO 00 


CM Cf> ON 
1^ N -> 











d 

PQ 


00 t^ Ov 10 

HH a\ a. vo 
10 Ti- 10 r^ 


00 Tl- CO 




>■* 








r^ 

^ 



C/3 

d 


-^ "* 

re Tl- Tj- vO 


00 Tt- vO 
NO "51- CO 




'-' 






d 

PQ 


« 10 Tl- ON 

fO -rt- 10 00 Q 
T^ rf 10 vO 5 


t^ r^ ID 




►H 






NO 

2 


in 

d 


10 vo a. Q 

On 10 00 00 
rO rj- n- l^ 


CI ON 
00 CO CO 











d 

PQ 


•1 <s ON 

10 1^ fD On Q 
10 10 vO t^ 


CI CI 00 

ON lO Tj- 











ts 

S 


c 

N 


^ ::^ > > 


>^ 



258 



THE BINARY STARS 



Since the zones are not of equal area, and since only the first 
lies wholly in the northern hemisphere, the fairest comparison 
is that afforded by the relative densities per square degree of 
double stars and all stars of the corresponding magnitudes. 
The double star densities were determined by dividing the 
figures in Table II by the number of square degrees in each 
zone area ; the figures were then reduced to a common standard 
by making the density in each column unity in Zone V, the 
Milky Way Zone. Seeliger has published corresponding data 
for all of the B. D. stars and the two sets of values are entered 
in Table III in the columns D. S. and B. D., respectively. It 
is clear that the density curves of double stars rise to sharper 
maxima in the Zone V than the corresponding curves of stars 
in general do. 

This fact is exhibited in a more striking manner if we tab- 
ulate, as in Table IV, the percentages of double stars in five 
areas, the Milky Way Zone, the 20° zone on either side of it 
and the areas north of +30° and south of —30° galactic 
latitude. 





TABLE IV 

Percentages of Double Stars 




Galactic Latitude 


B.D. Stars 
to 9.0 


Double Stars 


Percentage of 
Double Stars 


+90° to +30° 
+30 +10 
+ 10 —10 
-10 -30 
-30 -70 


26,948 

19,355 
26,477 

17,831 
10,368 


1,335 
996 

1,620 
966 
483 


4-95 
5.15 
6.13 
5.13 
4.66 



The increased percentage in Zone V must be accepted as 
real. Table III shows that stars of all magnitude classes par- 
ticipate in it, and an examination of my charts leads to the 
conclusion that it cannot be an observing effect, for some areas 
of all galactic latitudes were examined in summer, others in 
winter; the area north of +60° declination, examined almost 
exclusively with the twelve-inch, extends from —3° to +27° 



THE BINARY STARS 



259 









C3 







^ 


N rr> vO 00 6 


M Th w 






10 ro HH 


::! "O 






»-' 




° S 


> 








lA ^ 





d 


b* n N 


NO t^ vo 






Z 


►H HH CO Td- 10 


N >-> 






^ 


2 ^%:;,s 


On 00 






>«*• N 


f_, 







IH 




2 











Tf 


m 


d 


Tt- 00 ro -^ vO 


n ON M 






Z 


i-i Tj- 10 t^ Tf 


t^ (S •"• 






^ 


^^P5S 8 


M NO 


^ 


^ 


NO CNJ 


^ 







1-1 
















fO 


Tj- 


6 


10 10 M M .^ 


M C< On 






Z 


i-i vO vo •-» vD 


CO 














^ 




r< 00 10 


M 





NO « 














C4 


fO 


6 


(^ HH M fO fO 


•^ (N< C< 




Z 


CO VO fO rO 


rt- rj- w 






l-l HH (S 








».Q 


vO 00 rf Q 

HH fo Ti- t^ 5 


rj- 00 


M _ 





^ 


t^ W 


10 











' 


cs 


d 


rj- vO »0 •* 


10 NO C< 






z 


(S 10 vO -^ 


to HH 








" 






».Q 


« CO Tj- t^ 5 
M 


10 00 00 


M _ 





^^ 


NO W 
















' 


d 


10 vO "-I ^ 


On M Th 






Z 


CS 10 ON fO 00 


1-1 10 I-. 








" 








M l^ Q 
i-H CO CO 10 


10 NO 


fe 





^ 


10 M 


^ 











d 




d 


i-i n a\ >o vo 


r« t^ 00 




Z 


CO ON On t^ 


10 i^ I-. 






■^Q 


tH vO vD 00 P 
i-i M CO »0 


Tt f 00 


_ 





^ 


NO M 





10 











d 


d 


11 11 10 00 


t^ M CO 


Z 


Tt CO M 00 


t4- 00 CO 






l-l HH C< CO 


W 


1^ 
Q 







^ n: s > > 


>^ 



26o THE BINARY STARS 

galactic latitude and the areas of high galactic latitude, both 
north and south, were examined mainly with the thirty-six- 
inch refractor. We may therefore say that close visual double 
stars are relatively more numerous in the Milky Way than else- 
where in the sky. 

Since the stellar system quite certainly extends to a much 
greater distance in the plane of the Milky Way than at right 
angles to it, the natural inquiry is whether the increased double 
star density in the Milky Way is not merely a perspective 
effect. If this were the case it would seem that we might 
expect a relatively higher percentage of very close pairs in 
Zone V than of pairs of moderate separation. Table V, how- 
ever, in which the 5,400 pairs are grouped according to galactic 
latitude and angular separation, shows that the percentage 
increase toward Zone V is as great for stars with angular dis- 
tances of from 2" to 5" as for stars under i". 

Let us consider next the relation between the angular sep- 
aration and magnitude. This is shown in Table VI where the 
stars are arranged with these qualities as arguments. The 
sums of the numbers in the first two columns of the table are 
entered in the third, thence the numbers are given for uniform 
steps in angular distance to the final column. Every line of 
columns three to seven exhibits a marked increase in the num- 
ber of pairs as the angular distance diminishes. 

This is still more clearly brought out when the figures in 
these five columns are expressed as percentages of the total 
number of stars under 5" separation in each magnitude class. 
Now it is generally believed that stars of a given magnitude, 
for example from 8.6 to 9.0, are approximately at the same 
distance from us, on the average. Hence the observed increase 
in the number of pairs as the angular distance diminishes is 
not a mere perspective effect, but represents a real increase 
in the number of pairs as the orbital dimensions diminish. The 
uniformity of the percentages in each column in the lower 
division of the table and the fair agreement between the ratios 
of the figures in each line of the first two columns in the upper 
division both argue that the Lick Observatory survey was as 
thorough for the fainter stars as for the brighter ones. This 



THE BINARY STARS 



261 



was to be expected, for with the thirty-six-inch refractor a 
close and difficult double of magnitude eight to nine is as 
readily seen as a similar pair of brighter magnitude; in fact, 
I find that 123 of the 379 pairs with angular separation of 

TABLE VI 

The Distribution of Double Stars by Angular Distance 
and Magnitude 





to 


O.SI' 


0.00' 


1. 01' 


2.01' 


3.01' 


4.01' 


5.01' 


Dist. 




to 


to 


to 


to 


to 


to 


and 




0.50' 


I. 00' 


1 .00' 


2.00' 


3.00' 


4.00' 


5 00' 


over 


Mag. 


















^ 6.5 


75 


63 


138 


83 


62 


41 


31 


99 


6 . 6-7 . 


82 


52 


•134 


59 


42 


40 


21 


14 


7I-7-5 


103 


67 


170 


99 


64 


48 


31 


29 


7.6-8.0 


178 


132 


310 


164 


107 


85 


63 


26 


8.1-8.5 


310 


223 


533 


285 


173 


128 


III 


21 


8.6-9.0 


508 


413 


921 


532 


317 


217 


191 


II 


Totals 


1,256 


954 


2,206 


1,222 


765 


559 


448 


200 



Percentages 



^ 6.5 






39 


23 


17 


12 


9 


100 


6 . 6-7 . 






45 


20 


14 


14 


7 


100 


71-75 






41 


24 


16 


12 


7 


100 


7.6-8.0 






42 


22 


15 


12 


9 


100 


8.1-8.5 






43 


23 


14 


II 


9 


100 


8.6-9.0 






42 


25 


14 


10 


9 


100 


Totals 






42 


23 


15 


II 


9 


100 



0.25" or less, and 385 of the 877 pairs with angular separation 
between 0.26" and 0.50" discovered in that survey are of 
B. D. magnitude between 8.6 and 9.0. 

These statements are of significance also in connection with 
the figures exhibited in Table VII which shows the percentage 
of double stars of each magnitude class. The high percentage 
in the first line is partly due to the relatively large number of 



262 



THE BINARY STARS 



bright double stars wider than 5.0". If this number is reduced 
to the same order as that of the wider pairs for the other 
magnitude classes, the percentage becomes 9.1. 

TABLE VII 

Percentage of Double Stars hy Magnitude Classes 



Magnitude 


B.D. Stars 


Double Stars 


Percentage of 
Double Stars 


to 6.5 


4,120 


458 


II. T 


6.6-7.0 


3.887 


306 


7-9 


7.1-7-5 


6,054 


438 


7.2 


7.6-8.0 


11,168 


758 


6.8 


8.1-8.5 


22,898 


1,251 


5-5 


8.6-9.0 


52,852 


2,189 


4.1 



In view of what has been said above it is impossible to 
attribute the disparity in the percentage of double stars among 
the brighter and fainter stars to incompleteness of the data, 
and this becomes even more apparent when we recall that the 
distance limit adopted (5.0") is the same for the stars in the 
last magnitude class as for the stars as bright as 6.0. If the 
adopted limit were a function of the magnitude, like those 
suggested by Comstock and Pickering,^ and the value 5.0" 
were retained for stars of 6.0 magnitude, we would have prac- 
tically the same percentages in the first two lines of Table VII, 
and diminished values in the following ones, the final percen- 
tage (for stars 8.6 to 9.0) being only half its present 
value. 

The very high percentage of spectroscopic binary systems 
among stars as bright as 5.5 magnitude may or may not be 
significant in this connection, for we do not yet know whether 
that percentage will hold among the fainter stars, but on the 
evidence before us we may venture the suggestion that per- 
haps the stars of larger mass, and hence presumably greater 
luminosity, are the ones which have developed into binary 
systems. 

* See footnote, page 253. 



THE BINARY STARS 263 

The stars have been grouped in the foregoing tables without 
regard to their spectral class. We hiive now to see whether 
any of the results obtained vary with the star's spectrum. In 
June of the present year (191 7) Director E. C. Pickering and 
Miss Annie J. Cannon, of the Harvard College Observatory, 
generously permitted me to compare my list of double stars 
with the manuscript of the New Draper Catalogue of Stellar 
Spectra, which contains the spectral classification of over 
200,000 stars. It is due to their great kindness, for which I 
cannot adequately express my thanks, that I am able to present 
the results which follow. The comparison with the New 
Draper Catalogue has provided the spectral classification for 
3,919 of the 5,400 visual double stars in the northern sky. Of 
the remaining 1,481 stars only 15 are as bright as 8.0 magni- 
tude (and about half of those were inadvertently omitted 
from the list I took to Cambridge), 218 lie between 8.1 and 
8.5, and 1 ,248 between 8 . 6 and 9 . o magnitudes. Practically, 
then, the data as to spectral class are complete for stars to 
8.0, nearly complete for stars to 8.5, but less than half com- 
plete for the stars between 8.6 and 9.0 B. D. magnitude. 

The New Draper Catalogue is based upon objective prism 
spectra and therefore, in general, ddes not record separately 
the spectra of the components of such double stars as are 
considered here. The spectrum for these is either the spec- 
trum of the bright component, or, if the magnitudes are nearly 
equal, a blend of the spectra of the tw^o components. A very 
large number of the closer double stars have components 
which differ but little in brightness, and it has long been 
known that such pairs exhibit little or no color contrast. 
Indeed, Struve, in his discussion of the stars in the Mensurae 
Micrometricae showed that the color contrast between the 
components of a double star was a function of their difference 
of magnitude, and Lewis and other writers have extended his 
researches. In my own observing I have noted the components 
of hundreds of pairs of nearly equal magnitude as being of the 
same color. I may add that although my eyes are entirely 
normal as regards color perception I have never been able to 
see such violent contrasts in any pair of 'colored stars' as are 



266 



THE BINARY STARS 



scopic binaries evidently show a marked preference for the 
early spectral classes from Bo to Ao, the visual binaries for 
the classes from Ao to Ko. The maxima in certain of the sub- 
classes such as Ao and Ko are apparently real but probably 
without special significance. The most unexpected feature of 
the table is that the number of spectroscopic binaries of the 
M sub-classes is relatively much greater than that of the 
visual binaries. 

To study the relations of these groupings to those of the 
stars in general, I have combined the sub-classes into six larger 
classes, using the Harvard system, which is indicated by the 
heavy vertical rules in Table VIII, and have compared the 
results with Pickering's tabulation of the stars in the Revised 
Harvard Photometry. 

TABLE IX 

The Spectral Classes of the Binary Stars and of the Stars in the 
Revised Harvard Photometry 

Numbers 





B 


A 


F 


G 


K 


M 


All 


Vis. Bin. (all) 


157 


1,251 


532 


1,093 


837 


49 


3,919 


Vis. Bin. (to 6.5) 


59 


142 


65 


83 


93 


18 


460 


Spec. Bin. 


198 


161 


61 


71 


95 


19 


605 


R.H.P. Stars. 


822 


2,755 


1,097 


932 


2,531 


636 


8,773 



Percentages 



Vis. Bin. (all) 


4 


32 


14 


28 


21 


I 


100 


Vis. Bin. (to 6.5) 


13 


31 


14 


18 


20 


4 


100 


Spec. Bin. 


33 


26 


10 


12 


16 


3 


100 


R.H.P. Stars. 


9 


21 


13 


II 


29 


7 


100 



V. B. (all) -H.P. 
V.B. (6.5) -H.P. 
Sp. B. -H.P. 



-5 


+ 1 


+ 1 


+ 17 


-8 


-6 


+4 





+ 1 


+ 7 


-9 


-3 


+24 


-5 


-3 


+ I 


-13 


-4 



The numbers are not strictly homogeneous even for the Re- 
vised Harvard Photometry stars and the spectroscopic binaries, 



THE BINARY STARS 267 

but are sufficiently so to permit comparison. To make the data 
for the visual binaries more, directly comparable with the rest, 
I have given separately the numbers for the systems as bright 
as 6.5 magnitude B. D. The upper part of Table IX records 
the actual numbers of the stars of each category and of each 
spectral class; the lower part, the percentages and the excess 
or defect of each percentage when compared with that of the 
Revised Harvard Photometry stars. The figures in the last two 
lines may be affected to a certain degree by the lack of strict 
homogeneity in the data, but they certainly place beyond 
question the fact that spectroscopic binaries as bright as 5.5 
magnitude are far more numerous among Class B stars than 
among stars of other spectral classes, and that the visual 
binaries as bright as 6 . 5 magnitude are in excess among Class 
G stars and least numerous among stars of Classes K and M. 
The discovery of spectroscopic binary stars is quite inde- 
pendent of the distance of the system, for the displacement of 
the lines in the spectrum (disregarding variations in the physi- 
cal condition of the star) depends simply upon the velocity of 
the light source in the line of sight. The observed or angular 
separation of a visual binary, on the other hand, is a function 
of its distance from us, and it is necessary to consider whether 
this fact may not seriously influence the distribution shown 
in the table, for many different recent investigations have led 
astronomers to believe that stars of Classes F and G are, on 
the average, the nearest, those of Classes B and M the most 
remote. It is not easy to estimate the effect of this factor. 
Certainly the 5" limit imposed in collecting the data does not 
favor the Class G stars at the expense of those of Classes B 
and M, for it is clear that of two systems of the same linear 
dimensions the more distant one might fall within the limit 
whereas the nearer one might be excluded by it. The more 
distant pair might, it is true, fall below the resolving power 
of the telescope while the nearer pair would be well enough 
separated to be discovered, but this would not account for 
the fact that the percentage of visual binaries (to 6.5 magni- 
tude) is greater among the Class B stars than among those of 
Class F or Class K. Admitting the uncertainties arising from 



268 



THE BINARY STARS 



the unknown distances of the double stars and from errors 
in the classification, I am still of. the opinion that the last two 
lines of the table give a fair qualitative representation of the 
true distribution with respect to spectrum of the visual 
binaries among stars as bright as 6 . 5 magnitude, as well as of 
the spectroscopic binaries. 

The percentages for the visual binaries to 9.0 magnitude 
will quite certainly be modified when we can compare the 

TABLE X 

The Distribution of Double Stars by Spectral Class 
and Galactic Latitude 

















Un- 


B.D. 




B 


A 


F 


G 


K 


M 


known 


Stars 


I 


I 


II 


24 


51 


52 


2 


58 


4,276 


II 


I 


45 


50 


159 


121 


3 


117 


8,798 


III 


4 


87 


78 


170 


129 


5 


167 


13,874 


IV 


16 


260 


95 


186 


148 


8 


283 


19,355 


V 


109 


542 


118 


202 


168 


14 


467 


26,477 


VI 


22 


247 


lOI 


183 


122 


II 


280 


17,831 


VII 


3 


48 


48 


105 


74 


5 


84 


7,998 


VIII 


I 


II 


18 


37 


23 


I 


25 


2,370 


Sums 


157 


1,251 


532 


1,093 


837 


49 


1,481 


100,979 



Percentages 



I 


I 


2 


20 


25 


31 


14 


12 


16 


II 


I 


8 


42 


79 


72 


21 


25 


33 


III 


4 


16 


66 


84 


77 


36 


36 


52 


IV 


15 


48 


81 


92 


88 


57 


61 


73 


V 


100 


100 


100 


100 


100 


100 


100 


100 


VI 


20 


46 


86 


91 


73 


79 


60 


67 


VII 


3 


9 


41 


52 


44 


36 


18 


30 


VIII 


^ 


2 


15 


18 


14 


7 


5 


9 



3,919 double star spectra with the spectra of all the stars in 
the New Draper Catalogue. It may be pointed out also that 
the 9.0 magnitude limit may operate here to favor Classes 
B and A at the expense of Class K, for the former classes con- 



THE BINARY STARS 



269 



tain a high percentage of stars intrinsically bright, the latter 
a considerable percentage of 'dwarfs'. 

Table X is arranged to show the distribution of the visual 
binaries of different spectral classes with respect to the galactic 
plane. I have placed the systems for which the spectral class 
is unknown in the seventh column of the table and in the final 
column have added Seeliger's count of the B. D. stars to 9.0 
magnitude. 

Visual binaries of every spectral class increase in number as 
we approach the Milky Way, but only in Classes B, A and M 
(and in the systems of unknown spectrum) is the increase 



TABLE XI 

The Distribution of the Spectroscopic Binaries by Spectral Class 
and Galactic Latitude 





B 


A 


F 


G 


K 


M 


All 


I 





7 


I 


4 








12 


II 


3 


21 


5 


4 


3 


I 


37 


III 


6 


17 


6 


II 


7 


' 


48 


IV 


31 


27 


9 


4 


19 


3 


93 


V 


76 


44 


12 


25 


20 


9 


186 


VI 


63 


26 


17 


16 


20 


3 


145 


VII 


10 


II 


7 


6 


17 


I 


52 


VIII 


7 


8 


3 


I 


8 





27 


IX 


2 





I 





I 


I 


5 


Sums 


198 


161 


61 


71 


95 


19 


605 



greater than that shown by the B. D. stars to 9.0 as a whole. 
The fact brought out in Tables III and IV that visual double 
stars are relatively more numerous in the Milky Way than 
elsewhere in the sky must therefore be due to the systems of 
these classes, and especially, in view of the actual numbers, 
to those of Class A and of unknown spectral class. Possibly 
the latter are also largely of Class A. The strong concentra- 
tion of the Class M stars toward the Galaxy is due entirely to 



270 



THE BINARY STARS 



the thirteen stars of Sub-classes Ma and Mb. These fall in 
Zones IV, V and VI in the numbers 3, 7 and 3, respectively. 
The thirty-six stars of Sub-class K5 are as uniformly distrib- 
uted as those of Class K (G5 to K2). 

Table XI, exhibiting the distribution in galactic latitude of 
the 605 spectroscopic binaries, shows a similar increase in the 

TABLE XII 

The Distribution of Double Stars by Magnitude 
and Spectral Class 





B 


A 


F 


G 


K 


M 


B.D. 
Stars 


To 6 . mag. 
6.1-6.5 


38 
21 


85 

57 


38 
27 


53 
30 


61 

32 


39 
6 


4,120 


6.6-7.0 


15 


109 


51 


59 


72 


I 


3,887 


7.1-7-5 


20 


162 


64 


104 


82 


3 


6,054 


7.6-8.0 


30 


232 


126 


209 


137 


8 


11,168 


8.1-8.5 


21 


308 


128 


347 


220 


9 


22,898 


Sums 


145 


953 


434 


802 


604 


66 


48,127 



Numbers Expressing Relative Frequency 



To 6 . 5 mag. 
6.6-7.0 

71-7. 5 
7.6-8.0 
8.1-8.5 



87.4 
23.6 
20.4 
16.5 
5-5 



14. 1 
II. 4 
II. 9 

8-5 

5-5 



15-3 
12.8 
10.3 
II .0 

5-5 



12.9 
10.6 

7-8 
7.0 
5-5 



149 

4 
6 

9 

5 



(II. I) 
( 7.9) 
( 7-2) 
( 6.8) 
( 5.5) 



numbers in all spectral classes as we approach the Milky Way. 
In this case we are considering the entire sky and may there- 
fore bring out the effect of the Milky Way more strikingly by 
dividing the sky into three zones by lines at =^30° galactic 
latitude. The central zone contains half the sky area and it is 
seen that six-sevenths of the Class B systems, four-fifths of the 
Class M, but only five-eighths of those of each of the remaining 
classes fall within this half. Class B stars as a whole are rela- 
tively more numerous in the Milky Way than elsewhere in the 



THE BINARY STARS 271 

sky in a ratio as great as that shown by the spectroscopic 
binaries of this class, but stars of Class M are quite uniformly 
distributed over the sky. That so high a percentage both of 
the visual and of the spectroscopic binaries of this class are 
in or near the Milky Way is somewhat surprising. 

In Table VII it was shown that the percentage of double 
stars is higher among the bright than among the faint stars. 
To see whether this holds for all spectral classes, I have ar- 
ranged Table XII. 

The spectral class is known for less than half of the double 
stars of magnitude 8.6 to 9.0 and this group is therefore 
omitted from the table. The upper part of the table gives the 
actual numbers of pairs of the various classes; the lower part 
was formed as follows : The numbers in each row in the upper 
part (after combining the first two lines) were divided by the 
number of B. D. stars of corresponding magnitude. The 
figures in the six resulting columns were then made com- 
parable by introducing factors to make those in the bottom 
row all equal to 5.5, the percentage of double stars among 
stars of magnitude 8.1-8.6 in Table VII. 

The comparison of the first six columns in this part of the 
table with the figures in parenthesis in the last column, which 
are repeated from Table VII, shows that double stars of every 
spectral class except G contribute to the increase in the per- 
centage of pairs as the apparent brightness increases. The 
excess of double stars brighter than 6.5 magnitude in Classes 
B and M is very striking, but the actual numbers as given in 
the upper part of the table show that in the other magnitude 
grades double stars of Classes A, F and K are more effective 
in producing the observed progression. 

I have prepared tables similar to Table VI for the double 
stars of each spectral class, but for the sake of brevity I shall 
give only their summary. 

In forming the percentages I have, as in Table VI, omitted 
the pairs wider than 5 . o". It is apparent that the progression 
of numbers with decreasing angular distance begins to fall off 
with Class K, and is still less appreciable in Class M. The 
individual tables of which XIII is the summary indicate that 



272 



THE BINARY STARS 



the distribution by magnitude shown in Table VI holds in 
Classes A, F, and G, and also in Classes B and K, though in 
these it is less marked for the brighter stars. It does not hold 
for the small group of Class M stars. 



TABLE XIII 



The Distribution of 3,QIQ Visual Binaries by Spectral Class 
and Angular Separation 





o.oo' 
to 

I .00* 


I .01' 

to 
2.00' 


2.01' 

to 
300' 


3.01' 

to 
4.00' 


4.01' 

to 
500' 


over 
5.00' 


Class B 


60 


33 


29 


10 


9 


16 


A 


540 


283 


158 


122 


88 


60 


F 


222 


115 


55 


74 


44 


22 


G 


489 


236 


143 


106 


82 


37 


K 


273 


187 


144 


108 


78 


47 


M 


12 


10 


8 


5 


6 


8 



Percentages 



B 


43 


23 


21 


7 


6 


100 


A 


45 


24 


13 


10 


8 


100 


F 


43 


23 


II 


14 


9 


100 


G 


46 


22 


14 


10 


8 


100 


K 


34 


24 


18 


14 


10 


100 


M 


29 


25 


29 


12 


15 


100 



A large percentage of the visual binary stars have measur- 
able proper motions and are therefore among the stars rela- 
tively near us (say within a radius of 500 light-years) those 
in which rapid orbital motion has been observed being, on the 
average, the nearer ones. No investigation of their actual 
distribution in space, however, has so far been published ; nor 
has the spatial distribution of the spectroscopic binaries as a 
whole been investigated. 

Hertzsprung, however, has studied the distribution of the 
Cepheid binaries, and Russell and Shapley that of the eclipsing 



THE BINARY STARS 273 

binaries. In each case it is found that the majority of the 
stars investigated (sixty-eight Cepheids and ninety ecHpsing 
systems) lie within a region bounded by planes drawn parallel 
to, and comparatively close to the plane of the Milky Way, 
and that within this region they are scattered over a vast 
extent of space. In the case of the Cepheids the median plane 
of the region passes about 123 light-years south of the Sun, the 
mean distance of a system from this plane is 296 light-years, 
while their distances from a central point defined by a perpen- 
dicular from the Sun upon the median plane range from about 
500 to nearly 10,000 light-years. The median plane for the 
eclipsing binaries lies about 100 light-years south of the Sun, 
a considerable majority of the systems, including nearly all 
those of short period, lie within 500 light-years from this plane 
and all of the others within 1,000 light-years; whereas their 
distances parallel to the plane from a central point defined as 
before range up to at least 8,000 light-years provided that 
light suffers no absorption in its passage through space. As- 
suming what the authors consider "a plausible amount of 
absorption of light in space," the limits are cut down to 4,000 
light-years. 

It is almost unnecessary to add that both of these conclu- 
sions, however reasonable they may be regarded, rest upon 
more or less speculative hypotheses and are not to be con- 
sidered as definitely established. 



CHAPTER XI 

THE ORIGIN OF THE BINARY STARS 

Any theory of the origin of the binary stars must take 
account of the facts of observation which have been outHned 
in the three preceding chapters. Chief among these are the 
following : 

1. The great number of the binary systems. On the average, at 
least one star in eighteen of those as bright as 9 . o magnitude is a 
binary visible in our telescopes; at least one in every three or four of 
those as bright as 5 . 5 magnitude is a binary revealed by the spectro- 
graph. These are minimum values. Our knowledge of the fainter 
stars is very incomplete, but on the evidence before us we may safely 
say that one-third, probably two-fifths, of the stars are binary systems; 
some astronomers, indeed, are inclined to think that systems of the 
type of our solar system may be the exception rather than the rule. 

2. The considerable percentage of systems with three or more 
components. It is well within the truth to say that one in twenty of 
the known visual binaries has at least one additional member either 
visible in the telescope or made known by the spectrograph, and many 
systems are quadruple or still more complex. Evidence is also accum- 
ulating to the effect that many of the purely spectrographic systems 
are triple or multiple. It is a fact of undoubted significance that, as a 
rule, triple systems, whether visual or spectroscopic, consist of a close 
binary pair and a companion relatively distant. 

3. The close correlation between the length of period, or size of 
system, and the degree of ellipticity in the orbit. The visual binaries, 
with orbit periods to be reckoned in tens of years, have average eccen- 
tricity close to 0.5; the spectroscopic binaries, with periods to be 
counted in days or fractions of a day, have average eccentricity of 
about 0.2; and in each class the average eccentricity increases with 
the average length of the period. We have, then, an unbroken pro- 
gression or series of orbits from systems in which the two components 
revolve in a fraction of a day in circular orbits and practically in 
surface contact, to systems in which the components, separated by 
hundreds of times the distance from the Earth to the Sun, revolve in 



THE BINARY STARS 275 

highly elliptic orbits in periods of hundreds, perhaps thousands of 
years. 

4. The correlation between length of period and spectral type. 
The short period spectroscopic binaries are prevailingly of spectral 
Class B; the longer period spectroscopic binaries are usually of much 
later spectral class ; the visual binaries have few representatives among 
stars of Class B and are most numerous among stars of Classes F and G. 

5. The correlation between the relative brightness of the com- 
ponents and their relative mass. When the two components are of 
equal brightness they are of equal mass, so far as our investigations 
have been able to go; in other systems the brighter star is, almost 
without known exception, the more massive, but the range in mass is 
far smaller than the range in brightness. No system is known in 
which the two components have a mass-ratio as small as i /lo. 

6. The relatively great mass of a binary system, taking the Sun as 
standard. Among the spectroscopic binaries showing two spectra we 
have found only one {d Aquilae) which has a minimum mass value less 
than the Sun's; the other minimum values given in Table VIII of 
Chapter VIII, range from one and one-half to thirty times the solar 
mass. The average mass of the visual binaries for which the data are 
really determinate is nearly twice that of the Sun, and only one system 
is known (Krueger 60) in which the mass can with any probability be 
said to be as small as half that of the Sun. 

7. The spectroscopic binary of Class B is, on the average, fully 
three times as massive as the binary of later spectral class. We have 
no data for the mass of a visual binary of Class B. 

8. A few systems are exceptions to the general rules. For example, 
the short period spectroscopic binaries of early spectral class with very 
eccentric orbits; the long period spectroscopic binaries of late spectral 
class, and the long period visual binaries with nearly circular orbits; 
the occasional system in which the fainter star is apparently the more 
massive. 

This enumeration is not exhaustive; I have, for instance, 
omitted the facts, brought out in my statistical study of the 
visual binaries in the northern sky, that such systems are 
somewhat more numerous, relatively, among the stars of the 
Milky Way than among those remote from it; that the bi- 
naries which cause this apparent concentration are chiefly of 
spectral classes B, A, and M; that there are apparently more 
binaries, relatively, among the brighter stars than among the 



276 • THE BINARY STARS 

fainter ones, to 9.0 magnitude; for some of these relations 
may prove to have no physical significance but to arise from 
the fact that our material necessarily consists of selected stars. 
As it stands, however, the enumeration is ample to indicate 
the difficulties attending any attempt to formulate a theory of 
general applicability. 

If we consider a single binary system, it is conceivable that 
it might originate in any one of at least three different ways: 

1. Two stars, hitherto independent, might approach each other 
under such conditions that each would be swerved from its original 
path and forced to revolve with the other in orbits about their common 
center of gravity (Capture Theory). 

2. A single star, in its primal nebulous stage, or possibly even later, 
might divide into two, which would at first revolve in surface contact 
(Fission Theory). 

3. The material in the primal nebula might condense about two 
nuclei separated by distances of the order of those now existing be- 
tween the centers of the component stars (Independent NucleiTheory). 

No one of these theories is entirely satisfactory when we 
consider the binary stars as a whole, and this is sufficiently 
demonstrated by the fact that each one of the three has its 
advocates among able astronomers at the present time. Since 
I have no new theory to advance I shall content myself with 
an exposition of the principles involved in each of the three 
and its apparent accordance or discordance with the facts of 
observation. 

What I have called the capture theory appears to have 
been proposed originally by Dr. G. Johnston Stoney, in 1867. 
On May 15 of that year he presented a paper to the Royal 
Society of London On the Physical Constitution of the Sun and 
Stars, the major portion of which is concerned with the Sun. 
In Section 2 of Part II, however, he treats of multiple stars 
and, distinctly stating that his deductions are "of necessity, a 
speculation," argues that if two stars should be brought very 
close to each other one of three things would happen. The 
third is that "they would brush against one another, but not 
to the extent of preventing the stars from getting clear again." 
In this event his analysis indicates that the stars would there- 



THE BINARY STARS 277 

after move in elliptic orbits, but that their atmospheres would 
become engaged at each periastron passage. Since the atmos- 
phere of each star "is not a thing of uniform density," the 
resistance would take the form of forces acting, some tan- 
gentially, and some normally, to the stellar surfaces; the 
former would tend to reduce the periastron distance, the latter 
to increase it, and as one or the other dominated in a special 
case, the two stars would ultimately "fall into one another" or 
"gradually work themselves clear of one another." In the 
latter event a double star would result. 

Meanwhile, "the heat into which much of the vis viva of the 
two components has been converted will dilate both to an 
immense size, and thus enable the two stars gradually in suc- 
cessive perihelion passages to climb, as it were, to the great 
distance asunder, which we find in the few cases in which the 
final perihelion distance can be rudely estimated, a length 
comparable with the intervals between the more remote 
planets and the Sun. During this process, the ellipticity of 
the orbit is at each revolution decreasing; but if the stars 
succeed in getting nearly clear of one another's atmospheres 
before the whole ellipticity is exhausted, the atmospheres will 
begin to shrink in the intervals between two perihelion pas- 
sages more than they expand when the atmospheres get en- 
gaged, and will thus complete the separation of the two stars. 
When once this has taken place, a double star is permanently 
established." 

According to Stoney, then, the eccentricity of a system 
decreases and its major axis increases up to the time when the 
atmospheres of the two stars are completely disengaged even 
at perihelion passage. The most favorable case for the 
formation of a double star on this hypothesis is presented when 
the original stars are of equal mass; and since no double star 
can result unless the unequal pressure of the atmospheres in 
their grazing collision has imparted to at least one of the two 
"a swift motion of rotation," Stoney imagines that under 
special conditions the rotational motion might become so great 
as to exceed the cohesive strength of the star and there might 
"result two or more fragments spinning violently," ultimately 



278 THE BINARY STARS 



I 



leading to the formation of triple systems like that of 7" 
Andromedae. 

I have given Stoney's theory somewhat in detail because the 
fundamental principle in it, namely, that collisions of stars in 
various degrees, central, partial, or grazing, might produce the 
stellar systems as we now know them has frequently appeared 
in later speculations on cosmogony. Professor A. W. Bickerton 
has elaborated it into a complete theory of cosmic evolution; 
it was regarded favorably by Lord Kelvin (so far as the binary 
stars are concerned); it appears again in Arrhenius's book, 
Worlds in the Making, and has quite recently been presented, 
with apparent approval, by Moulton in his discussion of the 
origin of binary stars in the revised edition of his text-book, 
An Introduction to Astronomy. 

Moulton analyzes the special case of two stars each equal to 
the Sun in mass which, at a great distance apart, have a 
relative velocity that is zero. Let them approach each other, 
but assume that at the point of nearest approach they are as 
far apart as the Earth is from the Sun ; their relative velocity 
at that instant will be about thirty-seven miles per second. 
Now, if they encounter no resistance to their motion, they will 
simply swing round each other and then separate again, moving 
along parabolic paths. If, however, one or both encounter 
resistance "from outlying nebulous or planetesimal matter, or 
from collision with a planet" their velocities will be reduced 
and their orbits may be transformed into elongated ellipses. 
Moulton states that, in the assumed case, if the resistance 
reduces the velocity by 1/200 of its amount, or 0.185 miles 
per second, the stars will, after their nearest approach, recede 
to a distance of only 100 astronomical units, and that the 
reduction of velocity will generate only as much heat as the 
Sun radiates in about eight years, not enough to affect the 
stars seriously. The eccentricity of their orbits will be about 
0.98, their revolution period about 250 years. Collision with 
a planet comparable to Jupiter in mass would suffice to bring 
about the results described. 

If no subsequent collisions occur, the two stars will continue 
to move in very elongated ellipses about their common center 



THE BINARY STARS 279 

of gravity. "If there are subsequent collisions with other 
planets or with any other material in the vicinity of the stars, 
their points of nearest approach will not be appreciably 
changed unless the collisions are far from the perihelion point, 
their points of most remote recession will be diminished by 
each collision and the result is that both the period and the 
eccentricity of the orbit will be decreased as long as the process 
continues. If this is the correct theory of the origin of binary 
stars, those whose periods and eccentricities are small, are 
older on the average, at least as binary stars, than those whose 
periods and eccentricities are large . . . " 

It will be noted that the conditions under which two stars 
are imagined to be converted into a binary system are much 
more plausible in Moulton's development of the collision prin- 
ciple than in Stoney's, and that he introduces variables enough 
to make the theory competent to account for any particular 
form of binary star orbit actually observed ; we may vary the 
masses, the initial relative velocity, the degree of approach, 
the amount of resistance encountered. Moulton does not 
mention triple or quadruple systems in this connection; but 
to account for these it would seem necessary simply to imagine 
the encounter, under suitable conditions, of binaries relatively 
old with other single or binary stars. 

Objections from a philosophical point of view might easily 
be urged against this theory, but it will suffice to present a 
very serious one raised by the facts of observation, an objec- 
tion frankly acknowledged by Moulton and recognized also by 
Stoney — the very great number of the binary systems and the 
extreme rarity of near approaches or partial collisions of stars. 
Writing before the modern era of double star discovery had 
opened and when spectroscopic binaries were entirely un- 
known, Stoney says that if his theory is correct "we must 
conclude the sky to be peopled with countless hosts of dark 
bodies so numerous that those which have met with such 
collisions as to render them now visibly incandescent must be 
in comparison few indeed." Moulton, after pointing out the 
difficulty, says that its seriousness "depends upon the length 
of time the stars endure, about which nothing certain is 



28o THE BINARY STARS 

known." That is quite correct, but it is none the less possible 
as a matter of statistical calculation to estimate the probable 
frequency of stellar encounters, and thus to apply a numerical 
test to the probability of the theory. Such a calculation has 
recently been made (for a different purpose) by Professor 
J. H. Jeans.i 

Assume i,ooo million stars (perhaps ten times as many as 
are visible with our greatest telescopes) to be distributed in 
a space within a parallactic distance of o.ooi" from the Sun, 
a space which corresponds to a sphere having a radius of 3,260 
light-years. Their mean distance apart, if they are arranged 
in cubical piling, is then 10^^'^ centimeters, or about 330,000 
times the Earth's distance from the Sun. Under reasonable 
assumptions as to masses and velocities, Jeans finds that "a 
star is only likely to experience a non-transitory encounter 
about once in 4 X 10^^ (40 million million) years." To put it 
differently, only one star in every 4,000 will experience a non- 
transitory encounter in 10,000 million years. By a non- 
transitory encounter is meant one which will produce serious 
tidal deformations in one of the stars. If the stars are taken 
equal to the Sun in mass and given a relative velocity of forty 
kilometers per second, their separation at point of nearest 
approach must be not greater than the radius of Jupiter's 
orbit (about five astronomical units) to produce a lasting or 
non-transitory encounter. Transitory encounters, encounters 
in which the distance at point of nearest approach exceeds five 
astronomical units, will naturally be more frequent, but, with 
the data assumed, in 10,000 million years only about one star 
in three will have another star approach within 200 astronom- 
ical units of its center; that is, within seven times the distance 
of Neptune from the Sun! 

It is perhaps conceivable that a single encounter similar to 
the one Moulton describes might occur, but it is quite incon- 
ceivable that more than a very few should occur in any length 
of time that may reasonably be assigned for the age of our 
present stellar universe. Still more inconceivable is the sup- 

1 The Motion of Tidally-distorted Masses, with Special Reference to Theories of Cos- 
mogony. — Mem. R. A. S., 72, Part I, 1917- 



THE BINARY STARS 281 

position that a given star should have two or more encounters 
such as might produce triple or multiple systems. When we 
add the fact that the presence of a resisting medium adequate 
to convert hyperbolic or parabolic, into elliptic motion is a 
necessary condition that such encounters may result in binary 
systems, It appears that the theory would hardly be tenable 
even if we had but a few hundred systems to account for 
instead of the tens of thousands actually known. 

The two theories which remain to be considered both assume 
that the binary stars originated in nebulae, and in this respect 
they are in harmony with practically every theory of stellar 
evolution from the time of Kant, Herschel and Laplace to the 
present day. The early theories were based upon nebulae 
more or less spheroidal in form, like the one assumed by 
Laplace in his famous hypothesis for the origin of our solar 
system. The contradictions in this hypothesis to fundamental 
principles of mechanics and the discovery that the majority of 
the nebulae are spiral in form led most astronomers, some 
twenty years ago, to favor the spirals as the antecedents to 
stellar systems. Now, the trend of opinion is towards the 
theory that the spirals are independent or 'island' universes, 
and that the irregular gaseous nebulae, like the Great Nebula 
in Orion, are the most primitive forms of matter known to us. 

If a stellar system is to originate from a gaseous nebula,^ it 
is clear that whatever initial form we assign to the nebular 
mass, and whatever other qualities we assign to the matter 
composing it, we must conceive it to be endowed with gravi- 
tational power which, sooner or later, will produce motion in 
its particles. The ensuing evolutionary process is thus 
sketched by Campbell: 

"It will happen that there are regions of greater density, or 
nuclei, here and there throughout the structure which will act 
as centers of condensation, drawing surrounding materials 
into combination with them. The processes of growth from 

» The spectroscope shows that the spirals are, as a rule, not gaseous nebulae; instead of 
the bright-line spectrum of luminous gases, they show a continuous spectrum, with dark 
absorption lines. Only four spirals are known, which, in addition to the continuous spec- 
trum, show a few bright nebular lines. 



«82 THE BINARY STARS 

nuclei originally small to volumes and masses ultimately stu- 
pendous must be slow at first, relatively more rapid after the 
masses have grown to moderate dimensions and the supplies 
of outlying materials are still plentiful, and again slow after 
the supplies shall have been largely exhausted. By virtue of 
motions prevailing within the original nebular structure, or 
because of inrushing materials which strike the central masses, 
not centrally but obliquely, low rotations of the condensed 
nebulous masses will occur. Stupendous quantities of heat 
will be generated in the building-up process. This heat will 
radiate rapidly into space because the gaseous masses are 
highly rarified and their radiating surfaces are large in pro- 
portion to the masses. With loss of heat the nebulous masses 
will contract in volume and gradually assume forms more and 
more spherical. When the forms become approximately 
spherical, the first stage of stellar life may be said to have been 
reached." ^ 

If we start with the assumption that the binary systems as 
well as the single stars have developed from nebulae by some 
such process as Campbell has outlined, the question we have 
to consider is whether they were formed directly by conden- 
sation about separate nuclei, or whether, in a very early 
stage, they were single stars or spheroidal nebulae, dividing 
later under the stress of such internal forces as gravitation, 
radiation pressure and the forces of rotation, or under the 
strain of some external disrupting force. Each view has its 
advocates and its opponents, and so far as strict mathematical 
analysis goes, no definite answer has as yet been made. 

The behavior of a rotating homogeneous incompressible fluid 
mass in equilibrium and free from external disturbance has 
been made the subject of a series of brilliant researches by such 
mathematicians as Maclaurin, Jacobi, Poincare, and G. H. 
Darwin. It has been possible, under certain assumptions, to 
follow the transformations of form as the mass contracts under 
its own gravitation and heat radiation and to show that it 
passes from the initial sphere through a succession of spheroids, 

» W. W. Campbell, 'The Evolution of the Stars and the Formation of the Earth'. Scien- 
tific Monthly, September-Decamber, 191 S- 



THE BINARY STARS 283 

ellipsoids and pear-shaped figures till a stage is reached where 
it seems certain that the next transformation will be a rupture 
into two masses. The analysis becomes too complicated to 
permit this step to be demonstrated mathematically. 

The stars and the antecedent nebulae are not homogeneous 
nor incompressible, but it has been argued, first, I believe, by 
Dr. T. J. J. See, later by Darwin and others, that a nebula 
might none the less pass through a series of similar changes 
and ultimately form a double star. Darwin has shown that 
the two portions must have fairly comparable masses if the 
system, immediately after the rupture, is to be a stable one. 
Once formed, with components revolving in surface contact 
and in orbits practically circular, the agency of tidal forces is 
invoked to produce increase in the ellipticity of the orbit and 
in the length of the major axis. 

The potency of tidal friction, within limits to be noted later, 
is undeniable, and the whole theory is made exceedingly 
plausible by its apparent ability to explain many of the facts 
of observation. Thus, according to the classical theory of 
stellar evolution, the stars of spectral Classes Oe and B are the 
'young' stars; those of Classes A, F, G, and K, progressively 
older, the red stars of Class M, the 'old' stars, age being mea- 
sured not in duration of time but by the stage of development 
reached. Now, as we have seen, the spectroscopic binaries 
which, on the average, have the shortest periods and orbits 
most nearly circular, are young stars; spectroscopic binaries of 
long period and high eccentricity belong, on the average, to 
the later spectral classes; they are old stars. Visual binaries 
of known period are wanting among stars of very early or very 
late spectral class. Campbell, who favors the fission theory, 
argues that the binaries of very early type have components 
too close together to be separately visible, while those of very 
late spectral class have their components so far apart that the 
revolution periods are exceedingly long. He also thinks that, 
in many red binary pairs, the fainter component may have 
become so faint as to be no longer visible. 

Russell finds further support for the fission theory in the 
numerous triple and quadruple systems. His analysis leads 



284 THE BINARY STARS 

him to conclude that, with a distribution of masses such as is 
actually found in double star systems, if further division occurs, 
the resulting multiple system must consist of a (relatively) 
wide pair one or both of whose components are themselves 
more closely double, the distance separating the components 
of the closer pair or pairs being "less than about one-fifth that 
of the wider pair — usually much less." He finds support for 
his conclusions in his discussion of the distance ratios in the 
known triple and quadruple systems. 

The alternative theory, that double stars had their origin in 
separate nuclei in the parent nebula, was first suggested a 
century ago by Laplace. In Note VII to his Systbme du 
Monde he remarks: 

"Such groups (as the Pleiades) are a necessary result of the 
condensation of nebulae with several nuclei, for it is plain that 
the nebulous matter being constantly attracted by these dif- 
ferent nuclei must finally form a group of stars like the Pleiades. 
The condensation of nebulae with two nuclei will form stars in 
very close proximity, which will turn one around the other, 
similar to those double stars whose relative motions have 
already been determined." ^ 

Laplace's theory is quite generally accepted for those wide 
and irregular groups, like B Ononis, "for which the fission theory 
gives no explanation," and it is not without its adherents even 
for the usual type of binary system. The modern writer who 
has adopted it most explicitly is the very man who first for- 
mulated the fission theory, Dr. See. His discussion of the 
binary stars in the second volume of his Researches on the 
Evolution of the Stellar Systems is devoted more particularly to 
the development of the systems after their initial formation but 
on page 232 we find the statement, "It is evident . . . that 
the resulting mass-ratio in a system depends on the supply of 
nebulosity and the original nuclei already begun and slowly 
developing in the nebula while it was still of vast extent and 
great tenuity;" and, on page 584, the even more definite state- 
ment, "When a double star had been formed in the usual 

* See Essays in Astronomy, p. soi. (Edited by E. S. Holden; D. Appleton & Co., 
1900). 



THE BINARY STARS 285 

way (Italics mine) by the growth of separate centers in a widely 
diffused nebula ..." 

Moulton, as we have seen, favors the theory that binaries 
arose from entangling encounters of independent stars, but as 
between the two theories now under discussion, remarks 
" . . .we are led to believe that if binaries and multiple 
stars of several members have developed from nebulas, the 
nebulas must originally have had well-defined nuclei. The 
photographs of many nebulas support this conclusion." And 
Russell, while advocating the fission theory, admits that "The 
close pairs, almost in contact, revealed to us among the variable 
stars may be accounted for on either theory. The apparently 
universal fact that the components of a binary are comparable 
in mass is what might be expected as a consequence of the 
fission theory, but would probably have to be a postulate of 
the other." 

Now it is obvious that the separate-nuclei theory affords 
sufficient latitude for the explanation of any conceivable 
system. We may imagine the two nuclei to be so remote from 
each other originally that their initial approach will be prac- 
tically along the paths of parabolas and then invoke the action 
of a resisting medium to convert the motion first into that in 
an elongated ellipse and later into ellipses of ever smaller 
eccentricity and major axes; or we may imagine them placed 
so near each other that their relative motion is at first in small 
circular orbits which are afterward enlarged and made more 
eccentric by the action of tidal forces; and, in fact, this, 
crudely put, is the argument which See makes. That is, he 
considers that both of these opposing forces are actually effec- 
tive in producing the binary systems as we now see them, one 
or the other becoming dominant in a particular system. 

The fundamental objection to the separate nuclei theory is 
that we really do not explain anything; to use Moulton's 
words, "we only push by an assumption the problem of ex- 
plaining the binary systems a little farther back into the 
unknown." 

Russell's specific objection, based upon his study of the 
triple and quadruple systems is also of great force. Why 



286 THE BINARY STARS 

should these systems almost invariably consist of a compara- 
tively close binary pair attended by a third star or by another 
close binary pair at a distance relatively great? As Russell 
says, "Not only is there no apparent reason for it, but if we 
try to retrace in imagination the history of such a system, 
through stages of greater and greater diffusion as we penetrate 
farther into the past (keeping in mind that the moment of 
momentum of the whole system must remain constant), it is 
hard to form any idea of the history of the nuclei which will 
finally form a close and rapidly revolving pair, attended by a 
distant companion." 

All things considered, the theory which has most in its favor 
is the fission theory, though it must be admitted that there are 
very serious difficulties about accepting it unreservedly. 
These relate, first of all, to the possibility of the initial division 
of the parent nebula. I say nebula rather than star because 
Moulton's researches make it certain that binary stars in 
which the masses of the components are comparable and the 
periods such as we find in the visual systems must at the time 
of separation have had densities extremely small. This, in 
Moulton's words, "removes the chief support of the belief that 
there is any such thing as fission among the stars simply 
because of rapid rotation." Jeans comes to the same con- 
clusion as to the potency of rotation, but he finds that "gravi- 
tation also will tend toward separation," and that "a nebula 
can split into two parts under gravitation alone, the two 
nuclei being held apart by the pressure of the layer of gas 
which separates them, instead of by the so-called centrifugal 
force." 

But, granting the formation of a system by fission, and grant- 
ing also its stability, which, as Russell says "may well be 
accepted, on . . . physical grounds, unless direct mathematical 
evidence is produced to the contrary," we have still to ask how 
such a series of orbits as we actually observe could develop. 
Tidal friction is quite inadequate, as Moulton and also Russell 
have shown. At most, if the masses are equal, this force could 
increase the initial period to only about twice its value; if 
unequal, but comparable (and we must remember that the 



THE BINARY STARS 287 

greatest inequality of mass yet observed in a binary star is 
only about i :6), it might lengthen the period several fold, but 
certainly not from a few hours or days to many years, no 
matter how long the time. Similarly, while tidal friction can 
increase the eccentricity up to a certain point, it cannot trans- 
form a circular orbit into the highly eccentric orbits in which 
many of the visual, and even some of the spectroscopic binaries 
revolve. Campbell meets this difficulty with the statement, "if 
the tidal force is not competent to account for the observed 
facts . . . , some other separating force or forces must be 
found to supply the deficiency." No one can say that such 
forces do not exist; we can only say that as yet they remain 
unknown. 

The general conclusion of our discussion is that no theory of 
the origin and development of binary systems so far formu- 
lated seems to satisfy fully the facts of observation enum- 
erated at the beginning of the chapter. This need occasion 
no surprise, for binary star astronomy is still in its infancy. 
The entire history of the spectroscopic binary stars extends 
over less than thirty years, and these three decades have wit- 
nessed also some of the most important developments in the 
history of the visual binaries. Further, the researches which 
provide the mathematical bases upon which our theories must 
be built date for the most part from very recent years. 

We must remember also that the present is an age of dis- 
turbance and upheaval in nearly every branch of natural 
science. Physicists and chemists have been submitting the 
fundamental principles of their sciences to searching criticism 
with results that are well-nigh revolutionary; biologists of the 
highest ability are questioning some of the basic doctrines of 
the origin of species. It would be strange indeed if at this 
time it were possible to formulate a theory of the origin of 
the binary stars, or a general theory of stellar evolution, which 
would be really satisfactory, or even sufficiently satisfactory 
to meet with general acceptance. 

Thirty years ago, and even more recently, few seriously 
questioned the general outlines of the classical, or as one 
astronomer terms it, the conventional theory of stellar evolu- 



288 THE BINARY STARS 

tion. Stars were assumed to originate from nebulae somewhat 
after the manner described by Campbell in the passage which 
I have quoted; then, as the result of continuous radiation of 
heat, with the consequent contraction in volume and increase 
of density, to pass progressively through the stages marked 
for us by spectra of Classes B, A, F, G, K, and M, from 
white-hot stars to cooler yellow stars, orange stars and finally 
to red stars and thence to extinction. Facts of observation 
developed in more recent years seem to run counter to this 
order of development and raise difficulties so serious that a 
radically different order has been proposed and has won many 
adherents. This is the so-called two-branched order of devel- 
opment which has been most completely formulated by 
Russell. According to it the very young stars are red and of 
relatively low temperature; as they contract, they generate 
heat faster than it is radiated into space and the temperature 
rises. Consequently the stars pass through the spectral 
classes in the reverse order from that just given, and if they 
are sufficiently massive reach the white-hot state corresponding 
to a spectrum of Class B. From this point, with ever falling 
effective temperature, they pass through the spectral classes 
again in the normal order until they are once more red stars 
and finally become extinct. Less massive stars, the dwarfs, 
cannot reach the white-hot state, and turn at spectral Class 
A, or even F. 

Campbell, who with the majority of astronomers, favors 
the classical theory of stellar evolution, finds in the observed 
facts relating to the spectroscopic and visual binaries strong 
support for his views; Russell and others see in them confirma- 
tion of their theory. Personally, I am inclined to prefer the 
classical theory of general stellar evolution and the fission 
theory of the origin of binary stars, as working hypotheses, 
frankly admitting, however, that the observed facts offer dif- 
ficulties and objections which no means at present available 
can remove. 

REFERENCES 
Campbell. The Evolution of the Stars and the Formation of the 
Earth. Scientific Monthly, 191 5. 



THE BINARY STARS 289 

Darwin. The Genesis of Double Stars. Darwin and Modern Science^ 
PP- 543-564. Cambridge University Press, 19 10. 

. Presidential Address, British Association for the Advancement 

of Science. Report B. A. A. S., 1905, p. 3. 

Jeans. The Motion of Tidally-distorted Masses, with Special Refer- 
ence to Theories of Cosmogony. Memoirs Royal Astronomical 
Society, vol. 62, part I, 191 7. 

. On the Density of Algol Variables, Astrophysical Journal, 22^ 

93, 1905- 
. On the "Kinetic Theory" of Star Clusters. Monthly Notices, 



R. A. S., 74, 109, 1913. 

MouLTON. On Certain Relations among the Possible Changes in the 
Motions of Mutually Attracting Spheres when Disturbed by Tidal 
Interactions; and, Notes on the Possibility of Fission of a Con- 
tracting Rotating Fluid Mass. Publication 107, Carnegie Institution 
of Washington, pp. 77-160. 

. Introduction to Astronomy, revised edition, pp. 543-548. 

Russell. On the Origin of Binary Stars. Astrophysical Journal, 31, 
185, 1910. 

See. Die Entwickelung der Doppelsterne-System, Inaugural Disser- 
tation, 1892. 

. Researches on the Physical Constitution of the Heavenly 

Bodies. Astronomische Nachrichten, i6g, 321, 1905. 

. Researches on the Evolution of the Stellar Systems, vol. 2, chap. 



20, 1910. 
See also the References at the end of Chapter VIII. 



TABLE I^ 

List of Orbits of Visual Binary Stars 



No. 


Star 


Mag. 


Spec. 


I goo . 


JpOO . 


P 


T 


J 


S 3062 


6.5, 


7-5 


G5 


0^ I.O" 


+57° 53' 


y 

105 -55 


1836.07 


2 


22 


6 


8. 


7 


I 


A3 


3 


gm 


+79° 10' 


166.24 


1890.87 


3 


Ho 212 


5 


6. 


6 


4 


F 


30 


jm 


- 4° 9' 


6.88 


1905.27 


4 


^395 


6 


4. 


6 


5 


Ko 


32 


2in 


-25° 19' 


25.0 


1899-50 


5 


2 60 


3 


6, 


7 


9 


F8 


0^43 


Qm 


+57° 17' 


507 60 


1890.03 


6 


^513 


4 


7. 


7 


2 


A2 


1^53 


7m 


+70° 25' 


52.95 


1905.60 


7 


OS 38 


5 


4. 


6 


6 


A 


1^57 


8- 


+41° 51' 


550 


1892.0 


8 


S228 


6 


4, 


7 


3 


Fo 


2h 7 


6m 


+47° I' 


204.7 


1891.59 


9 


i3 524 


5 


6, 


6 


7 


Fo 


2'»47 


4- 


+37° 56' 


33-33 


1895.0 


10 


2518 


9 


4, 


10 


8 


Ao 


4*" 10 


7" 


- 7° 49' 


180.03 


1843.18 


II 


0279 


7 


5, 


9 


3 


Go 


14 


2m 


+ 16° 17' 


88.9 


1897.8 


12 


^883 


7 


9, 


7 


9 


F5 


4'' 45 


7™ 


+ 10° 54' 


16.61 


1907 03 


13 


Sirius 


— I 


6, 


8 


5 


A 


6»'40 


gm 


-16° 35' 


49.32 


1894.13 


14 


Castor 


2 


0, 


2 


8 


A 


7»>28 


2" 


+32° 6' 


346.82 


1969.82 


15 


Procyon 





5. 


13 


5 


F5 


34 


im 


+ 5° 29' 


39 


1886.5 


i6 


/3 loi 


5 


8. 


6 


4 


Go 


47 


2in 


-13° 38' 


23 -34 


1892.60 


17 


^581 


8 


7, 


8 


7 


G5 


7^58 


gm 


+ 12° 35' 


46.5 


1909.40 


i8 


f Cancri 


5 


6. 


6 


3 


Go 


8'' I 


s"- 


+ 17° 57' 


60 . 083 


1870.65 


19 


c Hydrae 


3 


7, 


5 


2 


F8 


8'' 41. 


5" 


+ 6° 47' 


15.3 


1900.97 


20 


23121 


7 


6, 


7 


9 


Ko 


9^ 12 


Om 


+29° 0' 


340 


1878.30 


21 


2 1356 


5 


9, 


6 


7 


Go 


23 


jm 


+ 9° 30' 


116.74 


1840.82 


22 


02208 


5 


o» 


5 


6 


A2 


45 


3" 


+54° 32' 


99.70 


1882.46 


23 


A.C. 5 


5 


8, 


6 


I 


A2 


9^47 


6m 


- 7° 38' 


72.76 


1880.54 


24 


^ Urs. Maj. 


4 


4. 


4 


9 


Go 


ii'^ 12 


gm 


+32° 6' 


59-81 


1875.76 


25 


02234 


7 


3, 


7 


7 


F5 


25 


4m 


+41° 50' 


77.0 


1880.10 


26 


02235 


5 


8, 


7 


I 


F5 


ii** 26 


7m 


+61° 38' 


71.9 


1909.0 


^ ^7 


7 Virg. 


3 


6, 


3 


7 


F 


I2»'36 


6m 


- 0° 54' 


182.30 


1836.42 


1^-28 


42 Com. Ber. 


5 


2, 


5 


2 


F5 


13'' 5 


jm 


+ 18'' 3' 


25-335 


1885.54 


29 


02269 


7 


2, 


7 


7 


A5 


28 


3m 


+35° 25' 


48.8 


1882.80 


30 


/3 6i2 


6 


3, 


6 


3 


F2 


13** 34 


6m 


+ 11° 15' 


23 05 


1907.22 


31 


/3 nil 


7 


4. 


7 


4 


Ao 


141^ 18 


5"^ 


+ 8° 54' 


44-32 


1920.4 


32 


a Centauri 





3. 


I 


7 


G5K5 


32 


gm 


-60° 25' 


78.83 


1875.68 


33 


2 1865 


4 


4. 


4 


8 


A2 


36 


4m 


+ 14° 9' 


130. 


1898.0 


34 


2 1879 


7 


8, 


8 


8 


F8 


41 


4m 


+ 10°. 5' 


238.0 


1868.30 


35 


02285 


7 


5. 


8 





F5 


41 


7m 


+42° 48' 


97-93 


1883.56 



^The orbits listed in this Table are discussed in Chapter VIII. 



TABLE I 

List of Orbits of Visual Binary Stars 



e 


a 


fi 


i 


w 


Angle 


Computer 


No. 


0.466 


1.44" 


37.4" 


±46.1° 


98.7° 


Inc. 


Doberck 


I 


0.40 


0.55^ 


154 -9° 


70.2° 


316.1° 


Dec. 


Glasenapp 


2 


0.725 


0.242" 


38.7° 


53.45° 


66.8° 


Inc. 


Aitken 


3 


0.171 


0.66" 


112.8° 


76.0° 


152.7° 


Inc. 


Aitken 


4 


0.522 


12.21*' 


99.2° 


31.6° 


88.9° 


Inc. 


Doberck 


5 


0.347 


0.61" 


81.4° 


35.9° 


6.2° 


Inc. 


See 


6 


0.82 


0.346" 


113.5° 


76.6° 


201.2° 


Dec. 


Hus3ey 


7 


0.41 


0.97' 


102.6° 


56.45° 


290.7° 


Inc. 


Rabe 


8 


0.60 


0.16" 


127.1° 


33.5° 


325.0° 


Dec. 


Aitken 


9 


0.134 


4-79" 


150.8° 


63.25° 


319.55° 


Dec. 


Doolittle 


10 


0.625 


0.57" 


66.0° 


56.2° 


129.8° 


Inc. 


Aitken 


II 


0.445 


0.19" 


34-2° 


9.35° 


190.0° 


Inc. 


Aitken 


12 


0.590 


7-55" 


43.27° 


+44-55° 


213-7° 


Dec. 


Lohse 


13 


0.44 


5.^56" 


33.9° 


63.6° 


277-6° 


Dec. 


Doberck 


14 


0.324 


405" 


150.7° 


14-2° 


36.8° 


Inc. 


L. Boss 


15 


0.75 


0.69" 


99.7°, 


79-8° 


74-65° 


Inc. 


Aitken 


16 


0.40 


0.53" 


116.5° 


59-4° 


282.0° 


Inc. 


Aitken 


17 


0.339 


0.856" 


Indet. 


0.0° 


183.55° 


Dec. 


Doberck 


18 


0.65 


0.23" 


104.4° 


+49-95° 


270.0° 


Inc. 


Aitken 


19 


0.33 


0.67" 


28.25° 


75 0° 


127.5° 


Inc. 


See 


20 


0.56 


0.844" 


144.3° 


66.2° 


122.1° 


Inc. 


Doberck 


21 


0.44 


0.32" 


186.5° 


14.6° 


342.2° 


Inc. 


Doberck 


22 


0.60 


0.41" 


197-95° 


37.1° 


46.9° 


Dec. 


Schroeter 


23 


0.411 


2.513' 


100.7° 


53.4° 


129.2° 


Dec. 


Norlund 


24 


0.302 


0.35" 


157-5° 


50.8° 


206.6° 


Inc. 


See 


25 


0.40 


0.78" 


78.5° 


43-6° 


135.0° 


Inc. 


Aitken 


26 


0.887 


3-74" 


40.4° 


29.9° 


260.4° 


Dec. 


Doberck 


27 


0.496 


0.674" 


11.2° 


90.0° 


278.7° 




Doberck 


28 


0.36 


0.325" 


46.2° 


71.3° 


32.6° 


Inc. 


See 


29 


0.52 


0.225" 


33.85° 


50.4° 


357.95° 


Inc. 


Aitken 


30 


0.15 


0.26" 


42.9° 


46.4° 


146.3° 


Inc. 


Aitken 


31 


0.512 


17.65" 


25 05° 


+ 79 04° 


52.35° 


Inc. 


Lohse 


32 


0.96 


0.62" 


129. ° 


39.7° 


129. ° 


Dec. 


Hertzsprung 


33 


0.70 


1.06" 


74-1° 


57.6° 


208.6° 


Dec. 


V. Biesbroeck 


34 


0.595 


0.34" 


Indet. 


0.0*' 


262.85° 


. Dec. 


V. Biesbroeck 


35 



TABLE I — {Continued) 



292 



THE BINARY STARS 



No. 


5/ar 


Mag. 


Spec. 


I goo . 


JQOO . 


P 


T 


36 


S 1888 


4.8, 


6.7 


G5 


14^ 46.8'° 


+ 19° 31' 


y 

159 -54 


1909.22 


37 


77 Cor. Bor. 


5 


6. 


6.1 


Go 


IS** 19. i'^ 


+30^ 39' 


41 56 


i892.26( 


38 


2 1938 


7 


2, 


7.8 


Ko 


20.7'° 


+37° 42' 


244-37 


1864.95 


39 


OS 298 


7 


4. 


7-7 


Ko 


32.5"^ 


+40° 8' 


56.653 


1882.86 


40 


7 Cor. Bor. 


4 


0, 


7.0 


Ao 


38.6- 


+26° 37' 


87.8 


1841.5 


41 


^ Scorpii 


4 


8, 


51 


F8 


15^58.9™ 


-11° 6' 


44.70 


1905.39 


42 


S 2026 


9 


Of 


9-5 


K5 


l6h II. I" 


+ 7° 37' 


242 . 10 


1907.64 


43 


S2055 


4 


0, 


6.1 


Ao 


25.9'" 


-f 2° 12' 


134- 


1811.5 


44 


r Herculis 


3 


0, 


6.5 


Go 


37 •5'" 


+31° 47' 


34 46 


1898.77 


45 


2 2107 


7 


0, 


8.5 


F5 


16^ 47.9™ 


+28° 50' 


221.95 


1896.64 


46 


/3416 


6 


0, 


8.5 


K5 


i7»» 12.2™ 


-34° 53' 


41.47 


1891.45 


47 


22173 


5 


9, 


6.2 


G 


25.2'° 


- 0° 59' 


46.0 


1915-2 


48 


/x Herculis 


10 


0, 


10.5 


Mb 


42.6°» 


+27° 47' 


43 23 


1880.20 


49 


T Ophiuchi 


5 


3, 


6.0 


F 


17'^ 57.6™ 


- 8° II' 


223.82 


1814.79 


'50 


70 Ophiuchi 


4 


I, 


6.1 


Ko 


i8h 0.4'" 


+ 2° 31' 


87.858 


1895.90 


51 


99 Herculis 


5 


2, 


10.5 


F8 


3-2™ 


+30° 33' 


53.51 


1887.84 


52 


A 88 


7 


2, 


7.2 


F8 


33-2'" 


- 3° 17' 


12.12 


1910.10 


53 


^648 


5 


2, 


8.7 


Go 


53.3'" 


+32° 46' 


45.85 


1914-15 


54 


r Sagittarii 


3 


4, 


3.6 


A2 


56.2"' 


-30° I' 


21.17 


1900.37 


' 55 


TCor.Austr. 


5 


0, 


50 


F8 


18^ 59-7'" 


-37° 12' 


124.65 


1878.46 


56 


Secchi 2 


8 


7, 


8.7 


G5 


i^h 7 g"* 


+38° 37' 


58. 


1894.0 


57 


f Sagittac 


5 


4. 


6.4 


A2 


44-5'" 


+ 18° 53' 


25.20 


1914-II 


58 


S 400 


7 


5. 


8.5 


G5 


20h 6.9™ 


+43° 39' 


81.04 


1888.23 


59 


^ Delph. 


4 


0, 


50 


F5 


32.9™ 


+ 14° 15' 


26.79 


1883.04 


60 


S2729 


6 


3, 


7.6 


F 


46. 1"™ 


- 6° 0' 


135.6 


1899.8 


61 


€ Equulei 


5 


.8, 


6.3 


F5 


2o'» 54 . 1™ 


+ 3° 55' 


97-4 


1873.5AP 


62 


5 Equulei 


5 


•3, 


5.4 


F5 


21^ 9.6™ 


+ 9° 36' 


5 70 


1901.35 


63 


T Cygni 


3 


• 8. 


8.0 


Fo 


10.8'" 


+37° 37' 


47.0 


1889.60 


64 


*c Pegasi 


5 


.0, 


51 


F5 


21^ 40.1™ 


+25° II' 


11-35 


1897.8 


65 


Krueger 60 


9 


• 3, 


10.8 


Ma 


22*» 24 . S"* 


+57° 12' 


54-9 


1929-3 


66 


/3 8o 


8 


•3, 


9-3 


G 


23'> 13.8™ 


+ 4° 52' 


95.2 


1905.0 


67 


/3 1266 


8 


•3, 


8.4 


F5 


25 -5™ 


+30° 17' 


36.0 


1911-35 


68 


85 Pegasi 


5 


.8, 


II. 


Go 


23*^56.9'" 


+26° 33' 


26.3 


1883.5 


I 


2 73 


6.1, 


6.7 


Ko 


o*> 49 . 6"" 


+23° 5' 


109.07 


1930.39 


2 


S 186 


7.0, 


7.0 


Go 


i»»50.7'° 


+ I°2I' 


136. 


1894.0 


3 


2483 


8.0, 


9-5 


G5 


3*^57-4" 


+39° 14' 


135-5 


1907.75 


4 


2 82 


7-9, 


9-5 


Go 


4^' 17. 1™ 


+ 14° 49' 


97-94 


1835-03 


5 


/3 552 


7.0, 


lO.O 


F5 


4»>46.2™ 


+ 13° 29' 


56.0 


1887.0 



THE BINARY STARS 



293 



e 


a 


i2 


i 


0) 


Angle 


Computer 


No. 


0.514 


4-97" 


170.8° 


39.3° 


337.0° 


Dec. 


Lohse 


36 


0.272 


0.89" 


25-25° 


58.5° 


218.0° 


Inc. 


Lohse 


37 


0.568 


1.44" 


169.4° 


44.8° 


26.1° 


Dec. 


Lohse 


38 


0.584 


0.88" 


2.1° 


65.85° 


21.9° 


Inc. 


Celoria 


39 


0.42 


0.73" 


111.0° 


84.2° 


99.2° 


Dec. 


Lewis 


40 


0.75 


0.72" 


27.2° 


29.1° 


343-6° 


Inc. 


Aitken 


41 


0.722 


1.78" 


10.2° 


45.7° 


162.2° 


Dec. 


MatzdorflF 


42 


0.68 


0.99" 


87.7° 


30.3° 


123-5° 


Inc. 


Lewis 


43 


0.458 


1.35" 


51.6° 


47.5° 


113.3° 


Dec. 


Comstock 


44 


0.522 


0.85" 


179.6° 


23-35° 


123.5° 


Inc. 


Rabe 


45 


0.552 


1.86" 


131.0° 


49-0° 


64.0° 


Dec. 


VoOte 


46 


0.18 


1.06" 


153.7° 


80.75° 


322.2° 


Dec. 


Aitken 


47 


0.20 


1.30" 


60.8° 


63.15° 


182.0° 


Inc. 


Aitken 


48 


0.534 


1.307" 


76.2° 


66.1° 


17.75° 


Inc. 


Doberck 


49 


0.499 


4-56" 


122.96° 


58.57° 


193.64° 


Dec. 


Lohse 


50 


0.763 


I. II" 


75.0° 


38.3° 


93.7° 


Inc. 


Lohse 


51 


0.273 


0.176" 


2.4° 


62.4° 


270.0° 


Dec. 


Aitken 


52 


0.305 


1.04" 


52.5° 


62.35° 


335-7° 


Dec. 


Aitken 


53 


0.185 


0.565" 


75.5° 


69-4° 


1.4° 


Dec. 


Aitken 


54 


0.332 


2.14" 


53.5° 


148.1° 


169.55° 


Dec. 


Doberck 


55 


0.50 


0.40" 


90. ° 


68. ° 


0. ° 


Dec. 


Russell 


56 


0.85 


0.32" 


4.6° 


78.1° 


65-0° 


Dec. 


V. Biesbroeck 


57 


0.46 


0.47" 


157.1° 


59.9° 


7.0° 


Dec. 


Burnham 


58 


0.35 


0.48" 


178.55° 


62.25° 


351.2° 


Inc. 


Aitken 


59 


0.35 


0.64" 


164.8° 


62.3° 


73.3° 


Inc. 


Aitken 


60 


0.72 


0.61" 


106.8° 


85.5° 


0.0° 


Dec. 


Russell 


61 


0.39 


0.27" 


21.0° 


81.0° 


164.5° 


Dec. 


Aitken 


62 


0.22 


0.91" 


149.8° 


42.7° 


105.5° 


Dec. 


Aitken 


63 


0.49 


0.29" 


109.2° 


77.5° 


106.1° 


Dec. 


Lewis 


64 


0.182 


2.86" 


113.6° 


39.0° 


161.0° 


Dec. 


Russell 


65 


0.77 


0.72" 


6.2° 


22.95° 


98.0° 


Inc. 


Aitken 


66 


0.24 


0.24" 


59.1° 


62.15° 


163.0° 


Dec. 


Aitken 


67 


0.46 


0.82" 


115.63° 


53-08° 


266.12° 


Inc. 


Bowyer and 
Furner 


68 


0.77 


0.94" 


109.5° 


39-2° 


71.1° 


Inc. 


Rabe 


I 


0.67 


1. 15" 


42.6° 


73.9° 


226.8° 


Inc. 


Lewis 


2 


0.79 


1.77" 


23.1° 


68.0° 


213.4° 


Dec. 


See 


3 


0.50 


0.94" 


39.8° 


59-8° 


68.1° 


Dec. 


Hussey 


4 


0.345 


0.53" 


4.1° 


56.7° 


90.2° 


Inc. 


See 


5 



TABLE I — {Continued) 



294 



THE BINARY STARS 



No. 


Star 


Mag. 


Spec. 


IQOO . 


IQOO.O 


P 


T 


6 


/3 794 


7.0, 


8.3 


F8 


11^ 48 . 3™ 


+74° 19' 


y 
42.0 


1914-25 


7 


2 2123 


7.7, 


7-7 


F5 


12^ I.O™ 


+69° 15' 


103.3 


1860.50 


8 


2 1639 


6.7. 


7-9 


A5 


19.4'° 


+26^ 8' 


180. 


1892.0 


9 


7 Centauri 


3.2, 


3-2 


A 


I2'» 36.0™ 


-48'' 25' 


211. 9 


1851.6 


10 


S 1768 


50, 


8.0 


Fo 


13*^330- 


+36° 48' 


220.0 


1866.5 


II 


2 1785 


7.6. 


8.0 


K2 


44-5™ 


+27° 29' 


199.2 


191304 


12 


^ 1270 


8.6, 


8.7 


F5 


13^58.8- 


+ 8° 58' 


32.5 


1912.2 


13 


S 1909 


5.3» 


6.2 


Go 


I5h 0.5- 


+48° 3' 


204.74 


1 793 48 


14 


7 Lupi 


3.6. 


3-8 


B3 


28.5" 


-40° 50' 


83.0 


1845.0 


15 


^ Urs. Min. 


7.0, 


8.0 


F2 


15^ 45 •I'" 


+80° 17' 


115- 


1902.7 


i6 


A 15 


8.4, 


8.7 


K5 


i6»>40.8°» 


+43° 40' 


109. 


1897.8 


17 


22438 


6.8, 


7-4 


A2 


i8'>55-8'^ 


+58° 5' 


233 


1882.50 


i8 


22525 


8.0, 


8.2 


F8 


19^ 22.5™ 


+27° 7' 


243 -9 


1887.09 


19 


02387 


7-2, 


8.2 


F5 


i9^46.o°> 


+35° 4' 


90.0 


1838.0 



I 



THE BINARY STARS 



295 



e 


a 


n 


i 


CO 


Angle 


Computer 


No. 


0.50 


0.345" 


109. 2« 


52.75'' 


225.0° 


Inc. 


Aitken 


6 


0.49 


0.32" 


56.9° 


49-7° 


79.1° 


Dec. 


See 


7 


0.70 


0.71" 


109.2° 


58.15 


18.1° 


Dec. 


Lewis 


8 


0.30 


1.92"' 


3.35° 


81.8° 


285.0° 


Dec. 


Doberck 


9 


0.87 


1. 12" 


31.4^ 


36.6° 


257.1° 


Dec. 


Lewis 


10 


0.40 


2.55" 


169.1° 


38.9° 


159- 1° 


Inc. 


V. Biesbroeck 


II 


0.42 


0.22" 


156.6° 


38.7° 


12.15° 


Inc. 


Aitken 


12 


0.445 


3.58' 


58.7° 


83.1 ° 


25.0° 


Inc. 


Doberck 


13 


0.70 


1. 10" 


93.5° 


90.=^° 


90. ±° 




See 


14 


0.80 


. 42 " 


16.3° 


62.25° 


165.0° 


Dec. 


Aitken 


15 


0.55 


1.24" 


1570° 


72.0° 


201.0° 


Dec. 


Evans 


16 


0.916 


0.53" 


Indet. 


0.0° 


178.3° 


Dec. 


See 


^ ' 


0.918 


0.952" 


9.0'' 


47.3° 


274-4° 


Dec. 


Doberck 


18 


0.60 


0.66" 


129.55° 


65 -75° 


284.7° 


Dec. 


Doberck 


19 



TABLE IV 

List of Orbits of Spectroscopic Binary Stars 

A star preceding the figures in the column T indicates that the time is measured from a pt 
ticular phase of the curve; from the epoch of light maximum, as in most of the Cepheid variablt 
from the epoch of light minimum as in f Ceminorum, or from an epoch of maximum or of zero relati 
radial velocity as in tt^ Orionis. 



No. 


Star 


Mag. 


Sp. 


IQOO . 


1900 . 


P 


T 

2410000 4 


I 

2 

3 
4 

5 


a Androm. 
Boss 82 
13 Ceti 
7r Androm. 
7r Cass. 


2.15 
5.16 

5-24 
4-44 
5.02 


A 

A2 

F 

B3 

A5 


oh 3 2^ 
22.9™ 
30. I"" 

31-5"^ 
37.8- 


+28° 32' 
+43° 51' 
- 4° 9' 
+33° 10' 
+46° 29' 


96^67 
3-956 
2.082- 
143.67- 
1.964+ 


7882 . 40 
8841.59c 
7484. 48i 
8564. I 4z 
9970.03; 


_ 6 

7 
8 


23 Cass, 
f Androm. 
V Androm. 


5-39 
4-30 
4.42 


B8 

K 

B3 


41. I" 

42.0™ 

0M4-3™ 


+74° 18' 
+23° 43' 
+40° 32' 


33-75 
17.767+ 
4.283- 


10577.41 
10024.88] 


9 

10 


a Urs. Min. 
<f> Persei 


Var. 
Var. 


F8 
Bp 


ih 22. 6"" 
37-4'" 


+88° 46' 
+50° 11' 


1 3-968+ 
1 ii.9yr. 
|i26.5 
1 63.25 


4890 . 04 

4509.0 

8290.42 

*8326.58 


II 

12 

13 
14 
15 


o Triang. 
oj Cass. 
/3 Arietis 
RZ Cass. 
TT Arietis 


3-58 
5-03 
2.72 
Var. 
5-30 


F5 
B8 

A5 
A 

B5 


47.4" 

48.2™ 

ii>49.i'« 

2h 39. 9^ 
2h ^3 7m 


+29° 6' 
+68° 12' 
+20° 19' 
+69° 13' 
+ 17° 3' 


1-737" 
69.92 
107.0 
I. 195+ 
3.854 


10793.821 

10426.02 

7632.0 

9449.732 

10370.259 


i6 
17 


/3 Persei 
o Persei 


Var. 
3-94 


B8 
Bi 


3h lym 

38.0"' 


+40° 34' 
+31° 58' 


1 2.867+ 
1 1 . 899yr. 
4-419+ 


♦2.264 
1901.855 
8217.924 


i8 
19 

20 


$ Persei 
X Tauri 

n Persei 


4-05 
Var. 

4.28 


Oe5 
B3 

G 


52.5'° 

3^55-1™ 
4^ 7.6^ 


+35° 30' 
+ 12° 12' 

+48° 9' 


6.951 

f 3-953" 
1 34-60 

284. 


8248 . 308 

7945.119 
7831.30 
10061.97 


21 


b Persei 


4-57 


A2 


10. 7"" 


+50° 3' 


1.527+ 


8956.166 


22 


1^4 Eridani 


3.59 


B9 


14.1'° 


-34° 2' 


5.010+ 


7562.266 


23 

24 

25 


63 Tauri 
02 Tauri 
d Tauri 


5.68 
3-62 
Var. 


A2 

A5 
A2 


17.7'" 
22.9™ 
30.2°^ 


+ 16° 32' 
+ 15° 39' 
+ 9° 57' 


8.425 
140.70 

3.571+ 


9819.0 

8054.723 

9734.992 



1 The orbits listed in this Table are discussed in Chapter VIII. 



TABLE II 

List of Orbits of Spectroscopic Binary Stars 

When two sets of elements are given, the lower figures for K and for a sin i relate to the second 
component if the two values for w differ by i8o°; if the lower value of P is an aliquot part of the 
upper value, the lower line of figures relates to an assumed secondary oscillation; in other cases the 
lower line relates to a third body definitely known or suspected in the system. 







K 


Vo 


a sin i 


mi'sitiH 








e 


km. 


km. 


million km. 




Computer 


No. 


CO 


{m + wi)2 


76.21° 


0.525 


30.75 


-11-53 


34 - 790 


0.180O 


Baker 


I 


233.2° 


0.152 


41.7 


+ 2.04 


2.240 




MissUdick 


2 


223.1° 


0.062 


34-35 


+ 10.5 


0.981 


0.0087 


Fox 


3 


350.53° 


0.573 


47.66 


+ 8.83 


77.200 


0.876 


Jordan 


4 


[45-1° 


0.009 


117.76 


+ 8.92 


3.180+1 




Harper 


5 


1225.1° 




117.40 




3.170+J 








269.71° 


0.405 


16.32 


— 4.06 


7.020 


0.0121 


Young 


6 


182.22" 


0.037 


25-69 


-29.83 


6.272 




Cannon 


7 




0.000 


{75-63 
[104.0 


-23.91 


4-454 1 
6.125 J 




Jordan 


8 


80.0° 


0.13 


3-04 




0.164 


O.OOOOI 


Miss Hobe 


9 


293.0° 


0.35 


2.98 


-14.8 


166.800 


. 0098 






347-29° 


0.428 


26.90 


+ 3.20 


42.298 


0.189 


Cannon 


10 


1257.14° 


0.107 


6.96 












135-56° 


0.121 


12.10 


-12.65 


0.287 




Harper 


II 


49-97° 


0.30 


29.64 


-24.82 


27.190 


0.164 


Young 


12 


19-7° 


0.88 


32.6 


- 0.6 


22.880 


0.042 


Ludendorff 


13 


154-7° 


0.052 


69.30 


-38.32 


1. 137 


0.0412 


Jordan 


14 


78.27° 


0.042 


24.77 


+ 7.81 


1.310 


0.0061 


Young 


15 


21.0° 


0.060 


41-3 


+ 3.40 


1.630 


0.021 


Curtiss 


16 




0.000 


9-4 


+ 4.1 


89 . 000 


. 060 








0.000 


fill. 92 
|i6o.o 


+ 18.46 


6.801 
9.717 


0.754 . 


Jordan 


17 


99.18° 


0.034 


7-87 


+ 15-40 


0.752 




Cannon 


18 


77.5° 


0.061 


56.18 


+ 12.95 


3-050 


0.073 1 


Schlesinger 


19 




0.000 


10.4 




4.950 


0.004 J 






301.99° 


. 062~ 


20.50 


+ 7-83 


80 . 000 




Cannon 


20 




151.75° 


0.22 


41.89 


+23-09 


0.838 




Cannon 


21 




331.75° 




152.5 




3.048 










124-33° 


0.014 


63-76 


+ 17-83 


4.393 




Paddock 


22 




[304-33° 




64.85 




4.468 








190.7° 


0.16 


36.5 


+36.4 


4.170 


0.041 


Jantzen 


23 


54-16° 


0.717 


27.12 


+42.59 


37-471 




Plaskett 


24 




0.000 


72.68 


+29.23 


3-570 


0.142 


Daniel 


25 



TABLE II — {Continued) 



298 



THE BINARY STARS 



No. 


Star 


Mag. 


sp. 


I goo . 


igoo.o 


P 


T 
24ioooo-\- 


26 


SZ Tauri 


Var. 


F8 


31.4™ 


+ 18° 20' 


3^.148+ 


IOOI6.I87 


27 


T Tauri 


4-33 


B5 


36.2™ 


+22'' 46' 


1-505" 


7892.50c 


28 


9 Camelop. 


4-38 


B 


44-1'" 


+66° 10' 


7.996- 


6480.35 


29 


7r4 Orionis 


3.78 


B3 


45-9™ 


+ 5° 26' 


9-519"+ 


8279.64 


30 


7r5 Orionis 


3-87 


B3 


49.o«> 


+ 2° 17' 


3 - 700+ 


*792i.64 


31 


7 Camelop. 


4-44 


A2 


4^49.3™ 


+53° 35' 


3.885- 


8281.176 


32 


14 Aurigae 


5-14 


A2 


5h 8.8« 


+32° 35' 


3-789" 


10802.715 


33 


a Aurigae 


0.21 


G 


9-3" 


+45° 54' 


104.022 


4899-5 


34 


^ Orionis 


0.34 


B8p 


9-7" 


- 8° 19' 


21.90 


7968 . 80 


35 


7; Orionis 


3-44 


Bi 


19.4'° 


- 2"^ 29' 


7.990- 


5720.821 


36 


^ Orionis 


5-79 


A 


21. 6« 


+ 3° 0' 


2.526" 


7916.36 


37 


X Aurigae 


4.88 


Bi 


26.2" 


+32° 7' 


655-16 


10629.78 


38 


b Orionis 


Var. 


B 


26.9™ 


- 0° 22' 


5-732+ 


♦9806.383 


39 


VV Orionis 


Var. 


B2 


28.5™ 


- 1° 14' 


1 1-485+ 
[120.0 


9836.021 














9819. 


40 


I Orionis 


2.87 


Oes 


30.5'" 


- 5° 59' 


29.136 


{7587-991 
17577-354 
















41 


f Tauri 


3.00 


B3 


31-7" 


+21° 5' 


138.0 


5769 -9 


42 


125 Tauri 


5-00 


B3 


33-5" 


+25° 50' 


27.864 


10471.607 


43 


Boss 1399 


5.00 


B3 


35-8- 


- 1° II' 


27.160 


7961.465 


44 


136 Tauri 


4-54 


A 


47. o« 


-^2r 35' 


5-969 


9362.52 


45 


/3 Aurigae 


Var. 


Ap 


52.2'« 


+44° 56' 


3.960+ 


7100.732 


46 


40 Aurigae 


5-31 


A 


5^59-6™ 


+38° 29' 


28.28 


10468.197 


47 


V Orionis 


4.40 


B2 


6h 1.9m 


+14° 47' 


131.26 


7975-16 


48 


Boss 1607 


5-50 


A 


18. 0™ 


+56° 20' 


9-944 


9341-776 


49 


RT Aurigae 


Var. 


G 


22.1'= 


+30° 34' 


3-728+ 


*3 - 423 


50 


7 Gemin. 


1-93 


A 


31-9™ 


+16° 29' 


2175-0 


101.6 


51 


f Gemin. 


Var. 


G 


6** 58 . 2« 


+20° 43' 


10.154 


*i-3i3 


52 


29 Can. Maj. 


4-90 


Oe 


7M4.5'" 


-24° 23' 


4-393+ 


7240.248 


53 


RCan. Maj. 


Var. 


F 


14-9" 


-16° 12' 


1-136+ 


7966.576 


54 


<r Puppis 


3-27 


K5 


26.1" 


-43° 6' 


258.0 


10419.0 


55 


ai Gemin. 


2.85 


A 


28.2"" 


+32° 6' 


2.928+ 


6828.057 


56 


az Gemin. 


1.99 


A 






9.219- 


6746 385 


57 


(7 Gemin. 


4.26 


K 


7»'37.o" 


+29° 7' 


19.605 


5824.019 


58 


c Volantis 


4.46 


B5 


8^ 7-6™ 


-68° 19' 


14.168+ 


9453 562 


59 


*c Hydrae 


3-48 


F8 


8»'4i.5" 


+ 6° 47' 


5588.0 


5375-0 


60 


K Cancri 


5-14 


B8 


^h 2 . 4« 


+ 11° 4' 


6.393 


6486.897 



THE BINARY STARS 



299 







K 


Vo 


a sin i 


mi'sinH 









e 


km. 


km. 


million km. 




Computer 


No. 


0) 


(m + w,)2 


76.66° 


0.24 


10.94 


- 3-15 


0.460 


0.0004 


Hayncs 


26 


242 . 88° 


0.087 


44-34 


+ 13.55 


0.914 




Parker 


27 


90.0° 


0.30 


9.0 


- 2.25 






Lee 


28 


152.27° 


0.027 


25 -93 


+23.27 


3.393 




Baker 


29 




0.000 


57-88 


+24.20 


2.945 




Lee 


30 


217-14° 


0.013 


35-15 


- 8.93 


1.877 




Harper 


31 


19.70° 


0.033 


21.56 


-10.74 


1. 123 




Harper 


32 


117-3" 


0.016"*" 


25 76 


+30.17 


36.848 1 




Reese 


33 


1297-3° 




32.45 




46.430 J 








254-76° 


0.296 


3-77 


+22.62 


1. 109 


O.OOOI 


Plaskett 


34 


42.3" 


0.016 


144-75 


+35.5 


15.901 


2.51 


Adams 


35 


184.71° 


0.065 


144.12 


+ 12.02 


4-995 


0.780 


Plaskett 


36 


135-52° 


0.171 


20.53 


- 0.15 


182.300 


0.56 


Young 


37 


359-33° 


0.098"*" 


100.96 


+20.15 


7.926 


0.605 


Curtiss 


38 




0.000 


132.37 


+20.77 


2.704 


0.358 ] 


Daniel 


39 


40.0° 


0.30 


130 




20 . 460 


J 






112.374° 


0.742" 
0.000 


113-68 
8.38 


+21.53 


30.560 1 
3-358 J 




Plaskett 


40 


9-8° 


0.180 


14-95 


+ 16.4 


27.900 


0.046 


Adams 


41 


335-0° 


0.55 


25-5 


+ 14-8 


8.160 




Cannon 


42 


87.02° 


0.765 


93-04 


+26.12 


22.380 




Harper 


43 


{191-44° 


0.022 


48.9 


-17. 1 


4. on ] 




Cannon 


44 


1 11-44° 




71.0 














0.000 


108.96 
1 1 1 . 04 


-18. 1 


5-934 
6.047 




Baker 


45 


178.40° 


0.556 


51-38 


+ 16.91 


16.550 ] 
20.140 J 




Young 


46 


-1.60° 




62.51 










1.58° 


0.599 


34-09 


+22.10 


49.270 




Harper 


47 


152.9° 


0.081 


67.19 


-13.74 


9.127 




Harper 


48 


95-016° 


0.368 


17.96 


+21.43 


0.856+ 




Duncan 


49 


16.31° 


0.298 


6.12 


-12.28 


174-720 




Harper 


50 


333-0° 


0.22 


13.2 


+ 6.8 


1.798 


0.0023 


Campbell 


51 


37.64° 


0.156 


218.44 


— 12.12 




4-58 


Harper 


52 


195-86° 


0.138 


28.64 


-39-70 


0.443 


0.0027 


Jordan 


53 


350.0° 


0.2 


18.0 


+88.5 


62 . 570 




Lunt 


54 


102.52° 


O.OI 


31.76 


- 0.98 


1.279 




Curtis 


55 


265.35° 


0.503 


13.56 


+ 6.20 


1-485 




Curtis 


56 


330.25° 


0.022 


34.21 


+45 - 80 


9.220 




Harper 


57 




0.000 


66.67 


+ 9-68 


12.999 


0.437 


Sanford 


58 


90.0° 


0.65 


8.45 


+36.78 


493 000 




Aitken 


59 


162.26° 


0.149 


67.8 


+26.3 


5.890 




Ichinohe 


60 



TABLE II — (Continued) 



300 



THE BINARY STARS 



No. 



6i 
62 

63 
64 

65 
66 

67 
68 
69 
70 

71 

72 
73 
74 

75 

76 
77 
78 
79 
80 

81 

82 

83 
84 

85 

86 

87 

88 
89 
90 



Star 



a Carinae 
Boss 2484 

K Velorum 
o Leonis 

3oHUrs.Maj, 

w Urs. Maj. 

/3 Urs. Maj. 
93 Leonis 
Boss 3182 
rj Virginis 

d2 Virginis 

e Urs. Maj. 
a2 Can. Ven. 
fi Urs. Maj. 

a Virginis 

Boss 351 1 
V Centauri 
h Centauri 
7j Bootis 
a Drac. 

d Bootis 

A Bootis 
5 Librae 
/3 Cor. Bor. 

a Cor. Bor. 

A 2 Serp. 
/3 Scorpii 

d Drac. 
<r Scorpii 
/3 Herculis 



Mag. 



356 
570 

2.63 
376 

4.92 
4.84 

2.44 

4-54 
512 
4.00 

524 

1.68 
2.90 
2.40 

1. 21 

4.96 

3-53 
4.76 
2.80 
364 

4.82 

483 
Var. 
3-72 

Var. 

5-37 
2.90 

4. II 
308 
2.81 



sp. 



B3 
A 

B3 

F5P 

A 
A 

A 

F8 
A5 
A 



Ap 
Ap 
Ap 

B2 
F 

B2 

B5 

G 

A 

F5 

G5 

A 

Fp 



B8 
Bi 

F8 
Bi 
K 



I goo . o 



8.4™ 
10.8™ 

19.0™ 
9^ 35 . 8"* 

ID** 16.9™ 
48.2™ 

10^55.8'° 
I !•> 42 . 8'° 

14.8'° 
40.6™ 

49.6^ 
I2h 51.4m 

13^ 19.9™ 



19.9" 



30 

43 

47 

13^49 

14^ I 



13 



is'* 23 



30 

40 
15^59 



i6»» o 
15 

25 



I goo . 



-58° 33' 
+47° 14' 

-54° 35' 
+ 10° 21' 

+66° 4' 
+43° 43' 

+56° 55' 
+20° 46' 
+78° 10' 

- 0° 7' 
+ 8° 13' 

+56° 30' 
+38° 52' 
+55° 27' 

-10° 38' 

+37° 42' 
-41° II' 
-31° 26' 
+ 18° 54' 
+64° 51' 

+25° 34' 

+35° 58' 

- 8° 7' 
+29° 27' 

+27° 3' 

- i°30' 
-19° 32' 

+58° 50' 
-25° 21' 
+21° 42' 



6^744 
[5.986 



116.65 
14.498 

11.583+ 
15.840+ 

0.312+ 
71.70 

1. 271 
71.9 

38.3 

4.i5yrs. 
550 
20.536+ 

4.014+ 

1.613- 
2.625+ 
6.927 
497.1 
51.38 

9 . 604+ 

211.95 
2.327 

[ 40.9 
1490.8 
17.36 

38.95 
6.828+ 

3.071- 
0.247- 
410.575 



THE BINARY STARS 



301 







K 


Vo 


a sin i 


mi^sinH 








e 


km 


km. 


million km. 




Computer 


No. 


CO 


(m + w,)2 


115-84° 


0.18 


21.5 


+23-3 


1.960 




Curtis 


61 


[355-2° 


0.504 


63.34 


-I3-II 


12.026 




Harper 


62 


li75.2° 




73.64 




13.981 








96.23° 


0.19 


46.5 


+21.9 


73.200 




Curtis 


63 




< 0.02 


[54 -0 
163. 1 


+27.07 


10.775 
12.571 


0.238 1 
0.378 J 


Plummer 


64 


171.9° 


0.381 


34.07 


— 0. 10 


5.020 




Schlesinger 


65 


11.95° 


0.264 


20.64 


-18.45 


4.336 




Parker 
Guthnick 


66 


315.4° 


0.124 


1.25 


-10.93 


2660 km. 


0.000 


and Prager 


67 


270.81° 


0.008 


26.54 


+ 0.17 


26.170 




Cannon 


68 




0.000 


63.2 


+ 0.3 


1. 104 




Lee 


69 


185.0° 


0.40 


27.6 


+ 2.2 


25.750 


0.126 


Harper 


70 


223.35° 


0.072 


40.99 


- 8.89 


21.530 




Cannon 


71 


43-35° 




80.0 












55.8° 


0.31 


3.5 


-12.9 


69.360 


. 0058 


LudendorfT 


72 


110.0° 


0.3 


21.5 


+ i.o 


1.500 




Belopolsky 


73 


103.96° 


0.535 


69.22 


- 964 


16.400 1 




Hadley 


74 


283.96° 




68.83 




16.400 J 








328.0° 


O.IO 


126. 1 


+ 1.6 


6.930 1 




Baker 


75 


148.0° 




207.8 




11.400 J 








201.5° 


0.067 


10.25 


+ 7.09 


0.227" 


. 0002 


Harper 


76 




0.000 


20.63 


+ 9.05 


0.745 


0.0024 


Wilson 


77 


147.23° 


0.23 


21.4 


+ 5.2 


1.984 


0.0065 


Paddock 


78 


315.20° 


0.236 


8.69 


- 0.23 


57-735 




Harper 


79 


19.07° 


0.384 


46.25 


-17.03 


30.173 




Harper 


80 


273.0° 


0.169 


68.40 


+ 9.80 


8.904 1 




Harper 


81 


1 93.0° 




72.05 




9.380 J 








223.42° 


0.54 


18.02 


+25.62 


44 . 000 


0.076 


Young 


82 


29.2° 


0.054 


76.5 


-45.0 


2.450 




Schlesinger 


83 


240.0° 


0.4 
0.000 


3.10 

2.4 


-21.28 


I . 600 1 




Cannon 


84 


312.2° 


0.387 


34-93 


+ 0.36 


7.671 


0.060 


Jordan 


85 


208.46° 


0.773 


50.52 


-11.63 


17.170 


0.133 


Jordan 


86 


20.09° 


0.270 


125.66 


— II.O 


11.360 1 




Daniel and 


S7 


200 . 09° 




197.0 




17.800 J 




Schlesinger 




126.11° 


0.014+ 


23.47 


- 8.36 


0.990 


0.004 


Curtis 


88 


110.0° 


0.05 


39.0 


+ 2.0 


0.129 


0.0014 


Selga 


89 


24.60° 


0.550" 


12.78 


-25.52 


60.280 


0.052 


Reese and 
Plummer 


90 



TABLE II — (Continued) 



302 



THE BINARY STARS 



No. 


Star 


Mag. 


SP- 


I goo . 


I goo . 


P 


T 
2410000 + 


91 
92 


€ Urs. Min. 
e Herculis 


4.40 
392 


G5 
A 


56.2'" 
16^56.5"" 


+82° 12' 
+31° 4' 


39^482 
4.024- 


8005 . 75 
8086.253 


93 


u Herculis 


Var. 


B3 


17^ 13.6™ 


+33° 12' 


2.051+ 


8125.80 


94 


Boss 4423 


4.61 


F 


21.3™ 


- 5° 0' 


26.274+ 


8411.524 


95 


^ Serp. 


364 


A5 


31.9™ 


-15° 20' 


2 . 292+ 


9209.618 


96 

97 

98 

99 
100 


CO Drac. 
X Sagittarii 
Y Ophiuchi 
W Sagittarii 
M Sagittarii 


4.87 
Var. 
Var. 
Var. 
4.01 


F5 
F8 
G 

F5 
B8p 


37-5" 
41.3'° 

47-3'^ 
I7b58.6°> 
i8»' 7.8^ 


+68° 48' 
-27° 48' 
- 6° 7' 
-29" 35' 
-21° 5' 


5.280- 

7.012- 

17. 121" 

7 • 595~ 
180.2 


7385-493 
6723-05 
*2.4i5 

*6.20 

4968.4 


lOI 

102 


Y Sagittarii 
108 Herculis 


Var. 

5-54 


G 
A 


15-5™ 
17.1'" 


-18° 54' 
+29° 48' 


5-773"^ 
5-5i5~ 


*4-46 
9551-742 


103 
104 
105 


X Drac. 
RX Herculis 
f 1 Lyrae 


369 
Var. 

4.29 


F8 

A 

F 


22. 9'" 
26.0°' 
41.3'" 


+72^41' 
+ 12° 32' 
+37° 30' 


281.8 
I . 779~ 
4.300- 


4864.3 
♦9658.588 
8109.722 


106 


/3 Lyrae 


Var. 


B2p 


46.4- 


+33° 15' 


12.919+ 


*9.867 


107 


50 Drac. 


5-37 


A 


49.6- 


+75° 19' 


4.120 


10293-93 


108 
109 
no 


61 Lyrae 
113 Herculis 


5-51 
4-56 
5.10 


B3 
G5 
B8 


50.2'° 
18'^ 50.5" 


+36^51' 
+22° 32' 
+ 10° 55' 


88.112 

245 -3 
1.302+ 


9220.727 
9805 . 
*8i57-502 


III 
112 
113 

114 
115 


18 Aquilae 
U Sagittae 
V Sagittarii 
2 Sagittae 

RR Lyrae 
a Aquilae 


Var. 
4-58 
6.03 

Var. 
517 


A 

B8p 
A 

F 
B8 


14.4'° 
i6.o°» 
19.8™ 

22.3™ 
34-3"^ 


+ 19° 26' 
-16° 8' 
+ 16° 45' 

+42° 36' 
+ 5° 10' 


3-381- 
137 -939 
7-390 

0.567- 
1.950+ 


8428.183 

9648 . 72 

10943-233 

♦0.508 
10054-331 


116 
117 
118 


SU Cygni 
77 Aquilae 
6 Aquilae 


Var. 
Var. 
3-37 


F5 

G 

A 


40.8°» 
19^ 47.4m 

20^^ 6.1=' 


+29° I' 
+ 0° 45' 
- 1° 7' 


3-844 

7.176 

17.124+ 


*2.5 

♦6.210 

8261.914 


119 
120 


/S Capricorni 
a Pavonis 


3 25 
2.12 


Gp 
B3 


15 ^"^ 

17.7'" 


-15° 6' 

-57° 3' 


1375-3 
11-753 


6035.0 
6379.90 



THE BINARY STARS 



303 







K 


Vo 


a sin i 


mi^sinH 


Computer 


No, 


O) 


& 


km. 


km. 


million kn. 


(m + m,)2 


359 46° 


0.011+ 


31-954 


-11.398 


17-346 




Plaskett 


91 


180.0° 


0.023 


70.39 


-24.03 


3-890 




Baker 


92 


0.0° 




112. 1 




6.200 








66.15° 


0.053 


99-50 


-21.16 


2.800 




Baker 


93 


246.15° 




253-0 




7.120 








14.48° 


0.491+ 


47-49 


+ 0.44 


14-950 




Parker 


94 


194.48° 




50.67 














0.000 


19-35 


-42-77 


0.610 




Young 


95 


333 76° 


O.OII" 


36.26 


-13.68 


2.632 




Turner 


96 


93-65° 


0.40 


15-2 


-13-50 


1-334 




Moore 


97 


201.7° 


0.163 


7.70 


- 5-10 


1.790 




Miss Udick 


98 


70.0° 


0.320 


19-5 


-28.6 


1.930 


0.005 


Curtiss 


99 


74-7° 


0.441 


645 


- 7-0 


143.500 




Ichinohe 


100 


32.0° 


0.16 


19.0 


+ 4-0 


1-485 




Duncan 


101 




0.000 


I 70.1 
.101.7 


—20.2 


5-320 1 
7.710 J 




Daniel and 
Jenkins 


102 


119.0° 


0.423 


17-95 


+32.38 


62 . 020 




Wright 


103 




0.000 


106.0 


-18.5 


2.590 




Shapley 


104 




0.000 


51-24 


-2597 


3-030 




Jordan 


105 


0.15° 


0.018 


184.40 


-20.95 


32.750 1 




Curtiss 


106 


180.15° 




75-0 




13-300 J 








(151-0° 


0.024 


75-77 


— 10. 1 


4.291 I 




Harper 


107 


1331-0° 




83.26 




4.716 J 








204.55° 


0.28 


33-68 


-25.85 


39 220 


0.309 


Jordan 


108 


169-5° 


0.12 


16.0 


-23.2 


53.580 


0.102 


Wilson 


109 




0.000 


27-59 


-18.65 


0.494 


0.0028 


Jordan 


no 


44-14° 


0.035 


66.45 


-19-13 


3.090 




Miss Fowler 


III 


28.6° 


0.087 


48.15 


+ 12. 1 


91 .010 


1.582 


Wilson 


112 


(332.6° 


0.05 


52.95 


+ 11. 


5-370 1 




Young 


"3 


1152.6° 




73-8 




7-490 J 








96.85° 


0.271 


22.2 


-68.7 


0.166+ 


0.00057 


Kiess 


114 




0.000 


(163-52 
I199.0 


- 5-0 


4-380 
5-340 J 




Jordan 


115 


(345-8°) 


0.2I=t 


25-=^ 


-33-4=*= 


1-350=^ 


0.0058^ 


Maddrill 


116 


68.91° 


0.489 


20.59 


— 14.16 


1-773 


0.0043 


Wright 


117 


I 14-9° 


0.681 


46.0 


-30.5 


7 930 




Baker 


118 


li94-9° 




63.0 




10.860 








124.0° 


0.44 


22.2 


-18.8 


377.000 




Merrill 


119 


224.80° 


O.OI 


7-25 


+ 2.0 


1. 170 




Curtis 


120 



TABLE II — {Continued) 



304 



THE BINARY STARS 



No. 


Star 


Afag. 


sp. 


igoo . 


I goo . 


P 


T 
2410000 + 


121 
122 


T Vulpec. 
57 Cygni 


Var. 
4.68 


F 
B3 


47.2- 
20^ 49 . 7™ 


+27° 52' 
+44° 0' 


4^436- 
2.855 


*3.678 
8554-770 


123 
124 
125 


/3 Cephei 

1 Pegasi 

2 Lacertae 


Var. 
396 
4.66 


Bi 
F5 
B5 


21'' 27.4"' 

22^^ 2 . 4™ 
16.9™ 


+70° 7' 
+24° 51' 
+46° 2' 


0.190+ 

10.213+ 

2.616+ 


9638.812 
4820 . 966 
8193.30 


126 
127 
128 
129 

130 


5 Cephei 
12 Lacertae 
77 Pegasi 
9 Androm. 
I H Cass. 


Var. 
5-18 
3.10 
5 90 
4.89 


G 

B2 

G 

A2 

B3 


25.4™ 

37.0- 

22^38.3™ 

23h 13 7m 

25-4" 


+57° 54' 
+39° 43' 
+29° 42' 
+41° 13' 
+58° 0' 


5.366+ 
0.193+ 
818.0 
3.219+ 
6.067 


7888.428 
10761.149 

5288.7 
11059-921 

8223 . 762 


131 
132 


X Androm. 
Boss 6142 


4.00 
6.05 


K 
Bp 


32.7m 
23^50.5- 


+45° 55' 
+56° 53' 


20.546 
13-435 


6683.46 
10800.634 


I 

2 

3 
4 

5 


a Persei 
a Orionis 
P Can. Maj. 
p Leonis 
a Scorpii 


1.90 
0.92 
1.99 

3-85 
1.22 


F5 
Ma 
Bi 
Bp 

Map 


3h 17 2^ 

5^49-8- 

6i»i8.3°> 

10** 27.5'" 

i6h 23.3m 


+49° 30' 
+ 7° 23' 
-17° 54' 
+ 9° 49' 
-26° 13' 


4.094- 
6 . yrs. 
0.25 
12.28 
5 . 8oyrs. 


7955-14 
6693. 

8749.603 
6673.582 



THE BINARY STARS 



305 



f Oi 




K 


Vo 


a sin i 


mx^sinH 







e 


km. 


km. 


million km. 




Computer 


No. 


(W+Wi)2 


104.03° 


0.440 


17.63 


- 1.39 


0.966 


0.0018 


Beal 


121 


1 45.0° 


0.137 


no. 4 


-16.2 


4.200 1 




Baker 


122 


1225.0° 




118. 8 




4.620 J 








2.63° 


0.040 


15.798 


-14.13 


0.041" 




Crump 


123 


251.81° 


0.0085 


47.99 


- 4.12 


6.740 




Curtis 


124 


ji8o.o° 


0.015 


80.3 


- 9.0 


2.890 1 




Baker 


125 


ij 0.0° 




98.8 




3.550 J 








85.385° 


0.484 


19.675 


-16.83 


1. 271 


0.0028 


Moore 


126 




0.000 


16.92 


-13.75 


0.045 


O.OOOI 


Young 


127 


5.605° 


0.155 


14.20 


+ 4-31 


157.800 




Crawford 


128 


40.57° 


. 0365 


73.56 


- 4.87 


3.240 


0.133 


Young 


129 


i 3.35° 


0.224 


59.06 


-14.78 


4.920 




Baker 


130 


301.0° 


0.086 


7.07 


+ 7.43 


1.990 




Burns 


131 


{339.56° 


0.105 


115. 5 


-26.7 


21.200 1 




Young 


132 


1159.56° 




167. 




30.700 J 








221.0° 


0.47 


0.93 


- 3-43 


0.046 


0.0000 


Hnatek 


I 


255.0° 


0.24 


2.45 


4-21.3 


70.000 


0.003 


Bottlinger 


2 




o.i=fc 


9.8± 


+33.7^ 


0.034- 




Albrecht 


3 


180. =t° 


0.5=^ 


10. =fc 


+41.1 






Schlesinger 


4 


289.0° 


0.20 


2.12 


- 3 09 






Halm 


5 



STAR LISTi 



A 88 84, 232 

A 1079 224 

A 1813 224 

A 2286 224 

Aldebaran 2 

Algol 28, 

. . 29, 167, 168, 247-250 
7 Andromedae . . 253, 278 
X Andromedae .... 246 
4 Andromedae .... 205 

a Aquilae 205 

7] Aquilae 221 

6 Aquilae 205, 206, 246, 275 
4 Aquarii 6 

Arcturus 2 

7 Arietis i 

a Aurigae 205 

/3 Aurigae . 29, 205, 244-246 
X Aurigae .... 198, 243 

RT Aurigae 221 

40 Aurigae 205 

d Bootis 205 

e Bootis 6, 264 

^ Bootis 216, 264 

Boss 2184 205 

Boss 4423 205 

Boss 6142 205 

9 Camelopardalis . . . 243 

f Cancri ... 6, 102, 216, 

. . . 223, 234, 253, 264 

^ Canis Majoris .... 193 



R Canis Majoris 
29 Canis Majoris 

f Capricorni . 

a Carinae . . 



190 
208 
232 
114 



rj Cassiopeiae 208, 

209, 216, 264 

7r Cassiopeiae 205 

Castor ... I, 2, 4, 9, 32, 
.... 65, 224, 240, 244 

Celaeno 46 

a Centauri . . . I, 2, 4, 36, 
114, 122, 124, 127, 209, 
. 216, 217, 226-230, 237 

/3 Cephei 195, 220 

8 Cephei 221 

13 Ceti 223 

82 Ceti 54 

42 Comae Berenices . 95, 232 
7] Coronae Borealis . . 209 
<T Coronae Borealis . . 193 

a Crucis i, 36 

SU Cygni 221, 223 

57 Cygni 205 

61 Cygni . I, 104, 226, 253 
W Delphini .... 181, 188 

50 Draconis 205 

Electra 46 

d Equulei . . 194, 209, 232 

e Equulei 54> 99 

7 Eridani . . . 114, 115, 205 

40 02Eridani 

209, 235, 236, 237, 265 

7 Geminorum 198 

K Geminorum 2 

f Geminorum . . . 163, 221 
(T Geminorum 2 

8 Herculis 6 

e Herculis .... 205, 246 
^ Herculis 102, 103, 209, 216 



1 No reference is made in tiiis list to the Tables of Orbits on pages 290-305. 



308 THE B I N A 

fjL Herculis 

209, 235, 236, 264, 265 
u Herculis .... 205, 206 

108 Herculis 205 

Hu 66 224 

Hu 91 224 

a Hydrae loi 

€ Hydrae loi, 198, 209, 210, 

... 216, 234, 253, 264 

Krueger 60 . . . 209, 216, 

230, 231, 275 

2 Lacertae 205 

7 Leonis 61 

o Leonis 205 

p Leonis 193 

/3 Lyrae 167, 168, 

. 201, 205, 206, 208, 250 
RR Lyrae . . . 195, 220, 221 

r Ophiuchi 6 

Y Ophiuchi 221 

70 Ophiuchi 

102, 209, 216, 250, 264 

a Orionis 193 

j8 Orionis loi 

5 Orionis 241, 242 

77 Orionis 205 

Orionis i, 4, 284 

^ Orionis 205 

02:351 224 

S 476 224 

K Pegasi . . . 223, 232, 250 
85 Pegasi . 209, 216, 232, 233 

a Persei 193 

b Persei 205 

o Persei 205 



R Y STARS 

Persei 193, 250 

Pleiades 45» 46 

Polaris 

237-240 
59, loi, 
228, 247 

237 
205 
xiii 
221 
221 
221 



. 207, 221, 223, 
Procyon .... 
. . . 209, 216, 
Proxima Centauri 
2 Sagittae 
V Sagittarii 
W Sagittarii 
X Sagittarii 
Y Sagittarii 
a Scorpii 
/3 Scorpii 
Sirius 
216, 



193 
205 
2, 59, loi, 209, 
. 228, 229, 233, 
..... 244, 247, 264 

S 2026 94 

S 3062 264 

X Tauri 205 

SZ Tauri 221 

80 Tauri 49, 50 

136 Tauri 205 

€ Ursae Majoris .... 198 

^ Ursae Majoris i, 

. . . 6, 7, 27-29, 63, 205 
^ Ursae Majoris 209, 216, 264 
W Ursae Majoris .... 190 
03 Ursae Majoris .... 205 
30H Ursae Majoris .... 246 
K Velorum .... 141, 146 
a Virginis 29, loi, 205, 244 
7 Virginis .... i, 4, 216 

d2 Virginis 205 

T Vulpeculae 221 



INDEX OF NAM ESI 



Adams 230, 231, 235, 264, 265 

Airy 18 

Albrecht 220 

Alter 104, 105 

Andr^ 106 

Argelander 247 

Arrhenius 278 

Astrand 74 

Auwers loi, 229 

Baker . . . 191, 224, 244-246 

Barnard 

vii, 24, 46, 47, 59, 230, 231 

Barr 201 

Bell 264 

Belopolsky . . . 116, 240, 248 

Bessel 

13, 19, 100, loi, 228, 229, 247 

Bickerton 278 

Bishop 18 

Bode 3, 4 

Bohlin 219 

Bond I, 63, 229 

Boothroyd 23 

Bosler 61 

Boss 215, 216, 233 

Bowyer 233 

Bradley i, 2, 14 

Burnham 17, 19-24, 26, 33-35, 
38,52,57,59,73, 193,219,229, 

230, 232, 253 

Burns 31, 246 

Campbell , . . vii, 30, 31, 44, 
72, 116, 117, 127, 132, 195, 
. . . 198, 204, 220, 222, 224, 

• . . 230, 237-239, 250, 251, 
281-283, 287, 288 

* No reference is made in this list to the 



Cannon, Miss A. J 

vii 112, 263, 264 

Chandler 247 

Clark, A. . . . 18, 22, 24, 235 

Clark, A. G loi, 229 

Clerke, Miss A. M 250 

Cogshall 23 

Comstock 45, 67, 94, 

. . . 102, 103, 233, 253, 262 

Cornu 123, 131 

Crossley 34 

Curtis . . . vii, 30, 31, 143, 

. . . 146, 149, 165, 240, 241 
Curtiss 128, 133, 

161, 165, 166, 222, 223, 225, 248 
Darwin . . 190, 282, 283, 289 
Dawes ... 18, 52, 56, 62, 67 
Dembowski . 18, 19, 22, 52, 67 

Ditscheiner 126 

Doberck I93-I95, 

218, 227, 240 

Doolittle 34, 35, 230 

Doppler 107 

Duncan 225 

Dunlop 36 

Eddington 210 

Elkin 46, 227 

Encke 66, 67 

Englemann 19 

Espin 23, 35, 253 

Everett, Miss A 218 

Feuille 2 

Fizeau 108 

Flammarion 33 

Fontenay i 

Fox 34, 63, 64 

Tables of Orbits on pages 290-305- 



310 



THE BINARY STARS 



Frost 127, 132, 243 

Furner 233 

Furness, Miss C 191 

Galileo 4 

Gill 40, 227 

Glasenapp 67, 74, 78-80, 84-86 

Gledhill I9» 34 

Goodericke 28, 247 

Gould 64 

Guthnick 195 

Hagen 191 

Hall 22, 33, 106 

Halley 2 

Hargrave 37 

Harper 162, 246 

Harting 168 

Hartmann 123, 129, I3i-I33» 242 
Henroteau . 160, 166, 250, 251 

Henry 64 

Herschel, J. 9, 10, 15, 21, 26, 33, 

35-37, 40, 66, 67, 71, 226, 253 
Herschel, W. . . . xiii, 3-13, 

IS, 21, 26, 33, 35, 40, 65, 223, 

236, 253, 281 

Hertzsprung . . . .64, 212-214 

220, 225, 239, 272 

Holden 24, 34 

Hooke I 

Hough .... 23, 33, 34, 253 

Howard 67, 106 

Howe 24 

Huggins 116 

Humboldt 228 

Hussey .... 17, 25, 26, 34, 

38, 56, 57, 72, 73, 209, 223, 253 

Huyghens i 

Innes . 36-39, 53, 61, 210, 237 

Jacobi 282 

Jeans 280, 286, 289 

Jonckheere .... 24, 35, 253 
Jordan 242 



Kant 281 

Kapteyn .... 233, 244, 249 

Kayser 132 

Keeler 132 

Kelvin 278 

Kepler 81 

King .... 154-157, 162, 165 

Kirchoff iii 

Klinkerfues 67, 105 

Knott 19 

Kowalsky 67, 74, 78-80,84-86, 95 

Kustner 24, 34 

Lacaille 226 

Lambert 2 

Laplace 281, 284 

Laplau-Janssen 225 

Lau 64 

Laves 160, 165 

Lee 243 

Lehmann-Filhes . 31, 142, 143, 

146, 148, 150, 160, 165, 209 

Leuschner 65, 106 

Lewis ... 3, 14, 16, 20, 34, 

56, 57, 60, 104, 193, 218, 263 

Lohse 227, 229 

Loud 225 

Ludendorff 

195, 201, 208, 222, 224, 225 

Lunt 193 

Maclaurin 282 

Madler ........ 19, 66 

Maury, Miss A. ... 29, 112 

Mayer, C 3, 4 

Mayer, T 223 

Michell 2, 8 

Moore vii, 31, 107 

Moulton 

vii, 96, 278-280, 285, 286, 289 

Newall 132 

Newton 65 

Nijland 165 



THE B I N A 

Opik 217 

Paddock 31, 162, 163, 164, 166 

Passement 11 

Peters loi, 229 

Pickering vii, i, 27, 64, 112, 168, 
239, 248, 253, 262, 263, 266 

Plummer 164, 165 

Poincare 282 

Pollock 37 

Poor 219 

Pound I, 2 

Ptolemy xiii 

Rambaut . . 31, 106, 148, 164 

Riccioli I 

Richaud i 

Roberts 168, 225, 227 

Rowland .... 123, 127, 128 

Russell, H. C 37 

Russell, H. N vii, 67, 

80, 99, 100, 148, 158-160, 164, 
. . . 165, 168-177, 181, 184, 
. . . 187, 190, 191, 212, 218, 
. . . 220, 225, 231, 272, 283, 

285, 286, 288, 289 

Safford 229 

Salet 61 

Savary 66, 67 

Schaeberle loi 

Scheiner 116, 132 

Schiaparelli .... 19, 33, 59 

Schlesinger ... 62, 63, 104, 

105, 138, 160, 161, 163, 166, 

. . . 195, 201, 204, 206, 224, 

231, 244, 245, 248 

Schonfeld 247 

Schorr 106 

Schuster 55 

Schwarzschild .... 67, 106, 

. . . 139, 149, 150-152, 165 

Secchi 19, 112 



R Y STARS 311 

See . . 23, 67, 73, 91, 106, 194, 

209, 218, 227, 283-285, 289 
Seeliger 67, 102, 

. . . 106, 234, 255, 258, 269 

Sellors 37 

Shapley 168, 188-191, 

. . . 216, 217, 225, 249, 272 

Smythe 18 

South 9-11, 21 

Stebbins. . . 190, 241, 247-249 

Steele, Miss H. B 211 

Stone 24 

Stoney 276-279 

Struve, F. G. W. . . 11-18,21, 

• • • 26, 33, 34, 36, 67, 99, 

232, 235, 253, 263 

Struve, H 62 

Struve, 17, 19, 21, 

... 25, 26, 34, 50, 232, 253 

Thiele, H 64 

Thiele, T. N. . . 67, 105, 106 

Tisserand 168 

Triimpler 46 

Turner 218 

Van Biesbroeck J215 

Villarceau 66 

Vogel . . 28, 29, 116, 132, 248 

Vodte 62 

Webb 22, 23 

Weersma .... 233, 244, 249 

Wendell 182 

Wilsing 148, 164, 165 

Wilson, J. M 19, 34 

Wilson, R. E 31 

Wright 31, 

... 117, 119, 132, 209, 227 
Young vii, 137, 138, 158,242,243 
Zurhellen 

. 139, 149-151, I53» 163-165 
Zwiers . . 67, 74, 80-83, 89, 95 



GENERAL INDEX 

Angstrom unit defined . . : 1 1 1 

Binary stars 

Capture theory of origin of 276 

Cepheid variables regarded as 219, 239 

Definitions of xiii 

Densities of 216 

Distribution of, by spectral class 265 

Distribution of, in space 272 

Fission theory of the origin of 282 

Masses of 202-216 

Multiple systems 223, 234, 235, 237 

Number of known orbits of 192 

Relations between period and eccentricity in 194 

Relations between period and spectral class in 198 

Separate nuclei theory of the origin of 284 

Statistical study of, in northern hemisphere 252-273 

Summary of the facts of observation of 274 

Systems of special interest 226 

Tables of the known orbits of 192, 290, 296 

See also. Double Stars, Eclipsing Binary Stars, Spectroscopic 
Binary Stars, Visual Binary Stars. 

Cepheid variables regarded as binary systems 219,239 

Color contrast in double stars 16, 263 

Diaphragms used in measuring double stars 58 

Dispersion curves. Construction and use of 123, 126 

Doppler-Fizeau principle 108 

Double stars 

Accidental errors in measures of 59 

Burnham's career as observer of 21 

Color contrast in 16, 263 

Conventions for measures and records of I5» 40 

Diaphragms used in measuring 58 

Distribution of, by angular distance and magnitude .... 260 
Distribution of, by galactic latitude and angular distance . . 259 



THE BINARY STARS 313 

Distribution of, by magnitude and galactic latitude .... 256 

Distribution of, by magnitude and spectral class 270 

Distribution of, by spectral class and angular distance . . . 272 

Distribution of, by spectral class and galactic latitude . . . 268 

Distribution of, in right ascension and declination 255 

Earliest discoveries of i 

Early speculations on the binary character of 2 

First collection of 3 

General catalogues of 34. 35 

Herschel's discoveries and theories 4 

Herschel (J.) and South's work on 10 

Lick Observatory survey for 24, 252 

Magnitude estimates of 52 

Method of measuring 48 

Modern period of work on 21 

Observing program for 53 

Percentage of , among stars to 9 . o magnitude 255 

Percentage of close pairs in certain catalogues of 26 

Photographic measures of 63 

Precautions to be observed in measuring 51 

Recent discoveries and measures of 23, 33 

Southern Hemisphere work on 36 

Struve (F. G. W.), and the Mensurae Micrometricae .... 11 

Struve (Otto), and the Pulkowa Catalogue 17 

Systematic errors of measures of 60, 62, 69 

Use of total reflecting prism in measuring 61 

Working definition of 252 

See also. Binary Stars, Visual Binary Stars. 

Driving clock. The first II 

Eclipsing binary stars 

Definition of 167 

Definitions of the orbit elements of 169 

Example of orbit computation of 181 

Formulae for density of 187 

Number of 168 

Russell's method of computing the orbits of 168-188 

Shapley's discussion of the known orbits of 188 

Evolution of a star 281 

Eye-pieces, Method of finding the magnifying power of ... . 58 

Gravitation, Universality of the law of 65 



314 THE BINARY STARS 

Kirchoff 's law iii 

Lick Observatory double star survey 24, 252 

Magnifying power of eye-pieces 58 

Magnitude estimates of double stars 52 

Micrometer 

Description of 40 

Methods of determining the screw-value of 42, 44 

Methods of determining the zero-point, or "parallel" .... 42 
Of the 40-inch Yerkes telescope, Barnard's measures of the 

screw-value of 46 

Multiple stars 223, 234 

New Draper Catalogue of Stellar Spectra 263 

Orbit of A 88 84 

Orbit of W Delphini 181,188 

Orbit of € Equulei 99 

Orbit of K Velorum 146 

Orbits of systems of special interest 226 

Parallel, Method of determining 42 

Photographic measures of double stars 63 

Proper motions of stars, Halley's discovery of the 2 

Radial velocity of a star 

Chapter on the, by J. H. Moore 107-133 

Components of the observed 127 

Correctness of measures of the, how tested 131 

Early determinations of the 116 

Modern precision of measurement of the 116 

Precautions to be taken in observing the 118 

See also: Spectrograms, Spectrograph^ Spectroscope. 

RectiHnear motion 103 

Resolving power of a spectroscope no 

Resolving power of a telescope 55 

Spectrogram of 0.2 Centauri, Measurement and reduction of . . 122 

Spectrograms 

Construction and use of dispersion curves in reducing . 123, 126 

Cornu-Hartmann formulae for 123 

Measurement of 122 

Principles governing choice of reduction methods for . ... 131 

Spectro-comparator method of reduction of 129 

Velocity-standard method of reduction of 128 



THE BINARY STARS 315 

Spectrograph 

Description of the Mills, of the Lick Observatory 117 

Loss of light in a stellar 120 

Relative advantages of refractor and reflector when used in 

connection with a stellar . 120 

Spectroscope 

Essential parts of a 109 

Resolving power of a no 

Spectroscopic Binary Stars 

American observatories, and work on 30, 32 

Conditions of the problem of computing orbits of 134 

Definitions of the orbit elements of 72, 137 

Differential formulae for correcting orbits of 160 

Discoveries of 30 

Distribution of, by spectral class and galactic latitude . . . 269 
Distribution of the longitude of periastron in orbits of . . . 201 

Earliest discoveries of i, 27 

Equation for radial velocity of 135 

Example to illustrate Lehmann-Filhes's orbit method for , 146 
First solution of the problem of determining orbits of ... 31 

King's orbit method for 154 

Lehmann-Filhes's orbit method for 142 

Lengthening of period in 246 

Masses of 205 

Number of known 30 

Number of known orbits of 31, 192 

Orbits of, with small eccentricity 164 

Period of revolution of, how determined 137 

References to orbit methods for 148, 160 

Relative masses in 205 

Russell's short orbit method for 158 

Schwarzschild's orbit method for 149 

Secondary oscillations in orbits of 162, 245 

Stationary H and K lines in 242 

Table of orbits of I93. 296 

Velocity curve for I39» 141 

Zurhellen's orbit methods for • 150 

See also Binary Stars 



3l6 THE BINARY STARS 

Spectroscopy, Fundamental principles of loi 

Spectrum, Definition of 109 

Star of smallest known mass 231 

Stellar spectra 

Classifications of 112 

Causes of line-displacements in 115 

Examples of 114 

Systematic errors in binary star measures 60-63, 69 

Telescope, Resolving power of a 55 

Tables of the orbits of spectroscopic binary stars 296 

Tables of the orbits of visual binary stars 290 

Total reflecting prism, use of in double star measures 61 

Variable proper motion loi, 228 

Visual binary stars 

Apparent ellipses of 67 

Construction of the apparent ellipse for, from the elements . 90 

Corrections to be applied to the measures of 68 

Definitions of the orbit elements of 72 

Early methods of determining the orbits of 66 

Examples of orbit computation 84, 99 

Formulae for differential corrections to orbits of ... 92, 94, 99 

Formulae for ephermerides for 73 

Formulae for orbits of, when i = 90° 95 

General equation for the apparent ellipse 70 

Glasenapp's modification of Kowalsky's orbit method for . . 78 

Interpolating curves for 71 

Kowalsky's orbit method for 74 

Masses of 209 

Methods of determining relative masses in ........ 214 

Minimum hypothetical parallax of 213 

Number of known orbits 192 

Parallax of, from spectrographic measures 209 

Parallelism of orbit planes of 218 

Relations between period, semi-major axis, and parallax of . 211 

Russell's statistical studies of 212 

Table of orbits of 192, 290 

With invisible component, or components 100 

Zwiers's orbit method for 80 



3S78 4 



86P 16 wn 



QB Aitken, Plobert Grant 

821 The binary stars 

A3 



Physical & 
Appfied Sci. 



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